Your PC can invent a new technique on his own
Taken from wikipedia's entry on the Southern Praying Mantis kung fu style:
The association of the term "Praying Mantis" with the style is also controversial. Each branch of the style offers a different explanation.
The traditions of the Chow Gar and Kwong Sai Jook Lum branches each maintain that their respective founders Chow Ah-Nam and Som Dot created their styles after witnessing a praying mantis fight and defeat a bird. Such inspiration is a recurring motif in the Chinese martial arts and can be found in the legends of Northern Praying Mantis, both White Crane styles, T'ai chi ch'uan, and Wing Chun.
If your rogue is observant enough - he may become inspired by some natural phenomenon, gaining insights about efficient and effective motions useful for offense, defense and maneuvering.
He doesn't need to be an acetic philosopher (though he may grow to become one as he advances) - he just needs the conviction and self-discipline to push him to go through the physical training required to hone his technique.
If your group's play style allows placing more emphasis on narrative at the cost of poorer combat optimization, you can even select to make the transition of your PC's reliance on his new martial skills gradual - continue wearing armor, combine armed attacks with unarmed strikes (not necessarily in the same round, though), etc. But, eventually your PC will start feeling that his armor is limiting him, preventing him from escaping blows that using his technique he can now avoid - then he may consider giving up on wearing armor. Same goes for unarmed strikes - have him hold on to his dagger, but occasionally use a flurry of blows(1) using a fist and an elbow of his other, empty hand - and let him realize that his hands have become faster and more accurate and than any weapon he is used to. This can lead him to relinquish the use of melee weapons, to use an interchanging combination of weapon and unarmed strikes or perhaps even seek "better" weapons which are more compatible with his developing abilities.
Bottom line - just because the "metagame concept" says your PC can now do something, doesn't automatically means he is aware of that and trusts himself enough to give up his old ways completely and immediately.
Or, he can find written records of a technique
Taken from wikipedia's entry on the Northern Praying Mantis kung fu style:
There are many legends surrounding the creation of Northern Praying Mantis boxing. One legend attributes the creation of Mantis fist to the Song Dynasty when Abbot Fu Ju [...] supposedly invited Wang Lang and seventeen other masters to come and improve the martial arts of Shaolin.
The Abbot recorded all of the techniques in a manual called the Mishou ("Secret Hands")
This manual supposedly disappeared until the Qianlong reign era,
The manual records Wang Lang "absorbed and equalized all previous techniques" learned from the 17 other masters.
Aside from self mastering a technique, your rogue may find a documentation of a certain technique - this can come instead of having him invent it from scratch, or as a latter supplement giving him more inspiration and advanced abilities.
At either case, you can probably work the narrative details with your GM so they won't clash terribly with the campaign settings (assuming that having a monk in it doesn't do that in the first place).
(1) Or whatever equivalent term used in 5e for that...
You should be able to play this
It is in no way optimal, but it's certainly possible to play this kind of character, and it will provide you with great roleplaying opportunities. A crippled fighter who uses what little Druidic magic he knows to get by where his physical form can no longer help is a great concept.
As you've already determined, having a low strength and a low dex, you are stuck with having terrible AC unless you wear heavy armor, which has a strength requirement unless it's a mithral armor. You will be able to use a shield alongside a club (heavy stick) for a bonus +2 AC, which means you'll be a fairly tanky character if you don't dump-stat your con as well for the theme.
You will also need to use your bonus action every minute to cast Shillelagh or you'll struggle to hit anything, but once you've done that, you're essentially just using a one-handed longsword, which is a perfectly viable build. You'll be weak to dispell magic and anti-magic zones, but in all honesty, if somebody is going to cast that to dispell your cantrip, they've wasted a spellslot.
For your once per day spell from Magic Initiate, you might consider Absorb Elements. Because you'll be dump-statting your dexterity, there's a decent chance you're going to fail dexterity saves as you are not proficient in those, and a well-timed Absorb Elements might well save you 10+ damage from a spell.
For your other cantrip options besides Shillelagh, you might want some way to 'close distance' or attack from a range, for those situations where your movement (especially when you're slowed down by armor) can't get you near enough to the enemy. With a decent wisdom modifier, your Thorn Whip would allow you pull enemies closer reasonably well. Gust might be useful to push enemies over edges from a distance or push unattended weapons around, allowing you to be more useful from a range than your slow speed would normally allow.
So in conclusion, is this going to be the strongest character at your table? Almost certainly not.
Is it going to be unplayable? I don't think so. I think that once you find Mithral Armor or find some other way to get a decent AC, your character will be perfectly viable in anything but a heavily min/maxed environment.
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.