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.
There is only one thing scarier than a farmer with her flock of chickens... a sainted farmer, worshiper of the colonel with a flock of undead, flesh-eating chickens, here to sell you delicious chicken flesh.
This idead inspired from this thread where:
... you are going to be playing a very different character than the cleric of a death god who has to build her (usually individually less powerful) undead from available corpses but gets a multitude of them
The thing about chicken-infested is... you'll never run out of corpses.
The theory:
The build: Commoner 1/Cleric 6 (suggestions courtesy of Cirdec)
The cleric, with access to the spell Blood Money via Dreamed Secrets can cast Animate Dead for so long as her Lesser Restoration spells hold out.
Given that you could be a normal cleric animating the dinosaurs you're fighting, the DM should be OK with these 3rd party feats, so long as you agree not to reanimate the corpses of the enemies: the blessings of The Colonel are to be reserved for the blessed beasts only.
Purchase a staff of command undead for 6,400 gp, basic enchanted armor and weapons, and keep the rest in onyx. When you get low, swap your burning skeleton swarms out for bloody skeleton swarms, which allows them to respawn. (They're less effective, but present a good way of conserving cash.)
Rulings needed:
- How do multiple fiery auras interact? How much damage should a swarm's firey aura do?
- How many HD does a burning chicken skeleton have? Does the swarm's HD "feel right?"
- Is it OK to take saint, in exchange for restricting its animate dead to only chicken you produce (providing you worship the undead chicken god named Colonel Five Spices?)
- What attacks and capabilities the "swarm of undead burning chicken" have.
A hypothetical burning undead chicken swarm:
A raven to raven swarm upgrades from one raven to 1000 ravens and uses the swarm rules for attacks. For undead chickens, since they're kinda-sorta flying, I would assert that an undead burning chicken swarm has 300 members (not a problem for you), which does "Swarm: 1d6" damage:
Swarm Attack: creatures with the swarm subtype don’t make standard melee attacks. Instead, they deal automatic damage to any creature whose space they occupy at the end of their move, with no attack roll needed.
Which significantly reduces the dice rolls needed.
Adding the burning skeleton template turns the attack into Swarm: 1d6+1d6 fire, some adjudication of how fiery death works when destroying swarm creatures, and turns the entire setup into a 4 HD (+1 from burning) creature. Of which you can have 3 (24 HD effective) on the table from your animate dead bucket, and 1 non-burning swarm from your command undead feat bucket (if you can figure out how to take it as a saint -- see if some adaptation of the grace rules is appropriate.)
As a swarm, it'll take half damage from piercing and slashing weapons, as well as DR/5 bludgeoning. I honestly don't know how to calculate how many HP the swarm has. but i would assume (4d8+4 = 22), which does not a huge amount of good when a t-rex chomps on them (as the t-rex has bite which counts as all three types for purposes of overcoming damage reduction), but will at the very least provide an excellent distraction, and the 3 swarms working together can take one down in 5 rounds (due to burning aura, the fact that every bite it takes does some damage to it) or less, assuming it is a little unlucky. Which means that you plus three swarms makes for a decent but not overwhelmingly powerful party member.
Nominal build:
Captain Cayenne
Human Commoner 1/Cleric 6
Motto: "Original... OR EXTRA CRISPY?!"
Happily, we worship the outer god, known as "The Colonel." His white beard and unspeakable chicken practices force us to acknowledge him as a Great Old One. Therefore, we have the simpler build.
Feats:
- Human Bonus: Quick Draw (needed to ... not just waste time)
- 1: Skill Focus: Diplomacy
- Traits: Chicken Infested, Magical Knack
- 3: Spell Focus (Necromancy)
- 5: Diabolical Negotiator (wisdom to diplomacy, increase more than two steps. All the better for selling "delicious, cooked from the inside, boneless chicken wings".)
- 7: Undead Master
With undead master, we have CL11 with regards to animate dead. Giving us 5 to 11 burning chicken skeleton swarms. In combat, stick close to someone so you can use bodyguard. (If you find yourself using it more frequently to better effect, trade out diabolical negotiator for combat reflexes).
An discussion on blood money
As per Cirdec's answer, the spell Blood Money may be used instead of the saint shenanigans. Unfortunately, Blood Money is magus 1, sor/wis 1, witch 1. And the best swarm-chicken casters are clerics. Therefore, we must class the cleric as an outer god.
Best Answer
I'll try to get the simpler issues done first - and the simplest one is Great Weapon Fighting. You should not take this. If you're using a spear, it is strictly inferior to the Duelling Fighting Style. See my answer here for the analysis. It's especially true for you because you want to tank, and for a quick and easy boost to your tankiness it's hard to go past a shield.
The next issue is how many levels of Paladin to take. This one gets a lot more complicated, but I'll start with the basics. When multiclassing in 5e, you want to take either 1 level or a number of levels that is a multiple of 4. This way, you don't lose any of the vitally important ability score increases.
The next thing to consider is what you actually want to get out of your Paladin levels, and what each number has to offer you. You also need to consider what Bard features you're missing out on by taking Paladin levels, but you've already got almost all the Bard features - from here on out your existing abilities mostly just get more powerful, rather than you gaining new things.
Paladin level 1: Gives basically nothing. You get the multiclassing proficiencies, Divine Sense, and Lay on Hands. Not a useful number.
Paladin level 4: Fighting Style, Divine Smite, Divine Health, and Channel Divinity. (And 1st-level Paladin spells, for what that's worth.) This is pretty good - a lot of the Paladin's core features for a minimal investment. If you're taking 4 levels, I'd recommend the Oath of Devotion, since (I think) it has the best Channel Divinity options.
Paladin level 8: All of the above, plus Extra Attack, Aura of Protection, and either Aura of Devotion or Aura of Warding. (Technically you could take Relentless Avenger, but it's not that great, especially for what you're trying to do.) Also 2nd-level Paladin spells, which is cool because one of them can be Find Steed, which is basically a class feature masquerading as a spell. This is a good place to finish taking Paladin levels because the next really good feature is at level 15. If you're going this far, the Oath of the Ancients starts looking as good as the Oath of Devotion, because Aura of Warding is so powerful.
I'd definitely recommend either 4 or 8 levels of Paladin if you're going to do this - 1 just seems like a waste of time. As for what order to take levels in, I'd actually suggest waiting till you've got your next ability score increase to do this, because doing it without War Caster might be a little awkward. Obviously, if you want to get better in melee sooner rather than later, you should start taking Paladin levels sooner rather than later, and you can work around the problems with casting your Bard spells with your hands full, it's just tricky.