The big problem you have here is MAD (Multiple Attribute Dependency). The bane of multiclass characters, MAD makes it difficult to pull off certain class combinations. As a Wizard, you're going to want as much Int as possible. As a Monk, you want to get both your Dex and your Wis up as high as you can. And since you've said you want to be a tank, you might even want to get a better Con score at some point.
You get 5 ability score increases over the course of your career - enough to max out your Int, then probably your Dex, leaving your Wis and Con scores exactly as they are now. (And that's assuming there's no feats you want.)
You can improve on this situation slightly: assuming I've done my maths correctly, it looks like you've used point-buy to come up with your ability score array. If you dump the Str and Cha scores back to 8, it gets you an extra 4 points which you can use to bring your Dex or Wis up to 16 (or your Con up to 14, but I wouldn't recommend that).
The way around the MAD problem is to pick classes which align slightly better, attribute-wise. For example, for a spellcaster Monk, picking a spellcasting class based on Wis (i.e. Druid or Cleric) would reduce your problems a long way. Unfortunately, there are no arcane spellcasting classes based on Wis.
Your next problem is that you've picked 5 skills that you want to be good at. The combination of class and background only gives you 4, and getting more requires a heavier investment than you can really afford, so you're not going to be able to get proficiency in all 5. The obvious one to ditch is Perception, because neither the Wizard class, the Monk class, or the Sage background give proficiency in it. That said, it's probably the most important skill in the game, so maybe you want to customize your background and drop something else. I'd suggest that one of the Int-based skills would be the best not to have proficiency in, since you'll have a good Int anyway so you'll still be ok at it.
Ignoring all that, let's assume that you're going ahead with a Wizard/Monk and address your specific questions.
Should I even multiclass, will it give anything at all? If yes, in what proportion? Will the Way of the Four Elements chosen or will other tradition give more benefits?
The 1st level of Monk gets you Unarmoured Defense and Martial Arts, which is enough to make it a good investment. Further levels in Monk take a lot of investment to get anything particularly worth having, so I'd recommend sticking with Monk 1/Wizard 19. If you do decide to take more Monk levels, make sure you do it in multiples of 4 so that you don't lose an ability score improvement. If you do take enough levels to get a Monk tradition, Way of the Four Elements is probably not a good choice. Your Wizard spellcasting will far outstrip the little bits of spellcasting it offers. The Way of Shadows likewise just gives some spells you have access to anyway. The Way of the Open Hand is probably your best bet.
Should I get a familiar?
What you have to understand is that familiars don't really boost your combat ability in any way. They offer great utility, as scouts/messengers, particularly if you pick something like the Hawk or Owl that's good at Perception. Regardless, it's not a big investment, just 5 gp and a single 1st-level spell (of which you have a minimum of 8 at Wiz 2), so I'd honestly recommend getting a familiar to anyone playing a Wizard.
What school to specialize in?
Abjuration is probably your best bet for helping you survive in melee. Necromancy is also good at this as long as you're killing things with spells. I went into a lot more detail about this in my answer here, so I'd suggest taking a look at that rather than me reproducing it here.
What spell choices can support the theme most?
There are now no Wizard spells that will directly boost your unarmed strikes, so your best bet for using a spell to increase your fighting abilities is probably just casting Haste on yourself. Other than that, your best bet for melee spellcasting is probably Vampiric Touch, particularly if you choose to specialize in Necromancy. Again, I went into a lot of detail about this here, so I'll keep self-linking instead of writing it all out again.
Feats or ability increases?
Ability score increases, definitely. You really need them, and there aren't any feats that you need as badly. War Caster is the usual choice for melee spellcasters, but you're not planning on holding any weapons or shields, so you don't really benefit from it much. Not to sound repetitive (it's actually a different answer this time), but I went into a lot of detail about which feats are good for a melee spellcaster in my answer here. But for you, I'd definitely suggest sticking to ability score increases.
Any magic items that can help?
Quite a few, actually. The ones you would want the most are the ones that would help you with your MAD problem: The Amulet of Health, the Headband of Intellect, Ioun Stones of Agility, Fortitude, Insight, or Intellect, Manuals of Bodily Health or Quickness of Action, and Tomes of Clear Thought or Understanding.
The other item that would be particularly useful for you specifically is the Bracers of Defense. Obviously, there are a lot of other items that are useful for everyone in general, or spellcasters in general. If you're playing Hoard of the Dragon Queen, there's an item called the Insignia of Claws that you definitely want to get if you can.
Based on simply damage output,
Pick Great Weapon Master
Assumptions before reaching 4th-level:
- You hit 65% of the time (+6 bonus to Attack Rolls)
- You are using a Greatsword (the best weapon to pick Great Weapon Fighting with)
- Disregard critical hits
- DPR is calculated as (%hit)(ave. weapon damage + str mod) or 65% x (8.33 + 4)
Currently, on a normal hit, you will do 8.0145 DPR, 10.82 DPR if made with Advantage (such as the effect of your Vow of Enmity).
If you pick Strength +2 at 4th-level,
Your attack bonus and damage rolls increase by 1, resulting to a DPR of 9.33 (70% x (8.33 + 5)), if with Advantage, 12.13.
If you pick Great Weapon Master,
Assuming you use the feat all the time, your attack bonus decreases by 5, and damage increases by 10, resulting in a DPR of 8.932 (40% x (8.33 + 4 + 10)). That's just slightly higher than what you started with.
However, when you have Advantage on the roll, your DPR increases to 14.29
TL;DR
On normal attacks, picking strength is slightly better than GWM. If made with Advantage, GWM is miles better.
Best Answer
I see that you want to stay a pure paladin, but I’m afraid I simply cannot recommend that. Feel free to ignore this answer if you really must, but a healer/tank paladin really wants to multiclass oracle. Pure paladin is just not very good at this. But paladin is a very good start to doing this.
Specifically, paladin is great for tanking thanks to divine grace. That is a fantastic ability. But paladin doesn’t have much more to offer for tanking or healing once you have that (if we’re being honest, it doesn’t offer much of anything once you have that—paladin is not a very good class).
But a paladin foundation can make for an extremely tough oracle. You have a bit more HP, and you have heavy armor proficiency, on top of divine grace adding your Charisma to every saving throw. Oracle then allows you to have strong spellcasting, also using your Charisma, and the mysteries can even find more uses for Charisma, such as the several mysteries that replace your Dexterity to AC with Charisma to AC (this is still limited by your armor, so if you go this way you may still want to use light armor even though paladin allows you to use heavy armor).
This so-called “oradin” is still a paladin. You would still need to adhere to the Code of Conduct, and also being an oracle would not change that. Even having only two levels of paladin, which is what I recommend, does not change that. What these things would change is that instead of martial prowess and damage-dealing, you would have much-superior spells—which is good, since it allows for better healing, better protection, and better support. In short, multiclassing oracle is how you make a more support-oriented paladin. The paladin features you miss out on are mostly best used for damage-dealing.
On which note, let me caution you against taking the title “healer” too literally. The cure wounds spell line is very, very inefficient. You may be called upon to cast those spells in emergencies, but the way things work out in Pathfinder, you are always better off protecting your allies from future harm (by buffing them, debuffing enemies, or applying some effect that separates one from the other) than trying to heal.
Instead, I recommend that you use a wand of cure light wounds (which you can use without a check since cure light wounds is on both the oracle and paladin spell lists) to keep everyone fit and ready in between combats, and only cast cure wounds spells in emergencies (probably just cure light wounds to get someone back into positive HP). Even with the life mystery available to potentially improve them, they just aren’t good spells.
The one exception here is the heal spell. That spell is incredible, and you should learn it ASAP. Once you know it, you will be able to heal huge chunks of HP at a time, while simultaneously removing a lot of negative conditions.
On that subject, spells that cure long-lasting adverse conditions, such as restoration, may be worth learning. They’re more expensive in wand form, and if they come up often enough, being able to heal them is very much worthwhile. In fact, it’s the most important kind of healing in the game.
But most of your spells should be about supporting, protecting, and enabling your party members.
To this end, I much prefer the sacred shield paladin archetype to the more apparently-appropriate hospitaler. The bastion of good ability is far more appropriate and useful to you than smite evil would be. The holy shield feature is also pretty solid, but I would recommend going for oracle before then.
For the oracle, the base version of the class is just fine for you; the warsighted oracle archetype is kind of interesting, and might recapture some of the martial ability lost when leaving the paladin class, but personally I would suggest that revelations are better than combat feats. Oddly enough, the black blood oracle archetype is also worth considering: the dark resilience revelation offers you greater protection against effects which could take you (the party’s healer) out of the running (leaving no one to heal you). It has some potentially-awkward fluff implications, but the black blood could also explain why you are taking a somewhat unusual approach to paladining than others might.
For mysteries and revelations,
The battle mystery has obvious appropriateness, and the battlecry revelation certainly does help out your allies. Battlefield clarity and war sight both help you stay up and available to support your allies, which can be important. And combat healer can actually kinda-sorta make cure wounds spells worthwhile, though it is so sharply limited in how much you can use it that I’m not sure I can actually recommend it.
Life is perhaps even more obvious. Unfortunately, since healing is inefficient, the life mystery is kind of meh, though it gets enough to be worth considering. Combat healer returns as an option, though again not as good an option as you might hope. Channel is likewise not all that great at healing, though it’s not awful. Enhanced cures could be amazing (if it’s allowed to apply to cure spells cast from a wand) or worthless (if not); ask your GM. Life link is OK but not amazing. Safe curing should be unnecessary. Spirit boost is meh until you get the heal spell, at which point is becomes quite good; I’d probably wait until I got that.
Nature has one of the Charisma-to-AC revelations, nature’s whispers, and it also has transcendental bond, which is great for organizing your party and even allows some otherwise-touch-range healing or buffing from a distance. Bonded mount also allows you to replicate some of your lost paladin class features, which is nice.
Time has the speed or slow time revelation, which is amazing for you since haste is not otherwise on the oracle spell list. That alone is a strong reason to consider this mystery. Temporal celerity can help you to get your buffs on allies at the start of combat, which is excellent, and rewind time can keep you going when you might otherwise be taken out of a fight. Time sight is also extremely useful to your party.
Personally, my recommendation is the time mystery. It hits a ton of high points for you.
Finally, if you are not playing in Golarion, you might consider the Divine Fighting Technique feat: the style associated with Desna in Golarion could be very nice for you, letting you use Charisma for attack and damage as well as AC and saving throws. Desna is a CG goddess, which is why this doesn’t work in Golarion, but if you aren’t there, maybe it could be associated with an LG deity.