[RPG] Are ability scores above 20 possible without magical means

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.

This sounds less like a combat issue than a roleplaying issue.

If the DM is having a wizard show up completely randomly and they are attacking you, then you need to discuss with the DM. How are they tied to a story, why are they attacking you, where are they supposedly coming from etc.

However, if the DM is having a Wizard show up completely at random but the players are the ones immediately attacking then that's a problem with you guys. Try talking to the wizard. See if you can defuse the situation (if there is one) without violence. You may also get better clues about their background and capabilities through conversation.

Essentially the easiest way to know how strong someone is is to ask them. You have a Paladin who's presumably got decent charisma. Try talking.

If conversation and role playing through this issue doesn't work then the most obvious spell to me is Detect Thoughts:

For the Duration, you can read the thoughts of certain creatures. When you cast the spell and as your action on each turn until the spell ends, you can focus your mind on any one creature that you can see within 30 feet of you. If the creature you choose has an Intelligence of 3 or lower or doesn't speak any Language, the creature is unaffected.

You initially learn the surface thoughts of the creature - what is most on its mind in that moment. As an action, you can either shift your attention to another creature's thoughts or attempt to probe deeper into the same creature's mind. If you probe deeper, the target must make a Wisdom saving throw. If it fails, you gain insight into its reasoning (if any), its emotional state, and something that looms large in its mind (such as something it worries over, loves, or hates). If it succeeds, the spell ends. Either way, the target knows that you are probing into its mind, and unless you shift your attention to another creature's thoughts, the creature can use its action on its turn to make an Intelligence check contested by your Intelligence check if it succeeds, the spell ends.