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.
Best Answer
I didn't find any references to costs of higher education, so I'm going to do a real world comparison.
Disclaimer
As it's well said in @NautArch's answer, this is not an accurate comparison, and I fully agree that a fantasy world is going to have a lot of nuances that will vary the prices of products, like guild based economy where competition doesn't exist and holds prices, for instance. But I still think it can serve as a reference to how much things cost in a more broader sense. The more you compare what things cost between them the more sense you can have of how much money (meaning coin) itself is worth.
It's impossible to find the real price of a product or service that exists only in fiction. There is no real way to compare it with something real moreso when the used coin itself is not real. This is not me saying "This is the accurate price". It's just "If X costs Y in our world, and Z in fantasy, you can compare it and have a clue of how much something is worth". Maybe the result is outrageous, maybe it makes sense to you. In the end it's fiction, so take this only as a way, or a tool to decide on the cost, nothing else. In the end prices are subjective. Economy is very complex, and in a roleplaying game, every product and every coin is put in circulation by the authority of the DM.
A little bit of math
A spiced apple cinnamon wine bottle, which is likely a cider, costs 5 silvers in Eberron. (Got the reference from here)
In Madrid a bottle of cider costs around 3€.
In Madrid higher education costs an average of 1,609€ a year.
Then we do a rule of three.
0.5 [gp cost of cider in Eberron] * 1,609 [Higher education in Madrid] / 3 [Cost of cider in Madrid] = The average cost of higher education in Eberron should go around 268.16 GP per year. This is more or less the cost of a coach with a horse, so it's pretty expensive.
Take in mind this is a rough approximate for an average university cost. Values could vary depending on market conditions. But I think it's a good measure of the basic cost, you could go up from there.
Probably magic schools in Eberron take a lot less students, which means increased prices, so you could get a more accurate price if you could know how many students does a magic school in Eberron have vs how many students does the average college or university have.
Since we got the cost of studying in Harvard from the answer user @Phillip gave us before (67,580$) we can use this method to do a similar comparison with the cider bottle in USA (around 5$ for a litre) giving us a total of 6,758gp a year.
DM controls the economy
But definitely, this is just another perspective and more of a thought experiment. @NautArch's answer is the most practical in regards to the game. The DM decides the value of money and controls all the economy, so the DM should decide.