Based on my 23 years as a professional live and studio guitarist (in order of appearance):
I swapped from 9s to 10s and it didn't sound very good (some buzz noise)
This is most likely because the action (string height) was set as low as possible for the 9's, and when you put on 10's the strings were too low to clear some frets. Highly unlikely the added tension from those 10's was strong enough to immediately pull your neck out of alignment. Instead of adjusting the truss rod, I would have raised the bridge saddles.
so I tightened the truss rod ... I realised the strings were too high
Upon realizing this, I would have "loosened" the truss rod slightly back in the opposite direction (undo what you did to raise them up).
when detuning the high E string, it snapped
Since detuning loosens tension, I predict your string broke due to rubbing across a sharp edge on your bridge saddle or string tree. Where did it snap? Take a file and smooth the metal surface(s) to prevent future strings from breaking.
I put on a new set (I only had 11s left)
You don't have to replace all of the strings when one breaks. You can buy individual [E] strings, and it's also totally OK to mix and match if you're in a bind (like at a gig). In fact, Ernie Ball even sells "custom" mixed gauge sets.
Should I leave it like at, or is there a way to adjust that, or will it eventually snap or keep going out of tune so should I put on a new set?
The main reason for wrapping them a certain way is to keep alignment with the nut/neck path. Generally you want the strings to continue a straight (or straight-ish) line of sight instead of pulling to the side. But it doesn't matter a huge deal how you wrap strings around your tuners. Yes some wrap methods will hold better than others but eventually if they wrap around the pegs enough times they'll hold. A bigger factor for intonation is whether they can pass/slide through the nut easily. If your nut was carved for 9's and you now play 11's then it's possible the nut grooves need to be widened (very slightly) to allow for the strings to glide easily. Otherwise, when you bend (either with fingers or "whammy" tremolo bar) you may experience intonation issues.
I'll clip the strings once I'm done with the setup
Purely cosmetic. Clip them, wrap them in a cool way, just don't cut yourself on the sharp edges.
I lowered the bridge and I feel like the action is on a decent level now, but I still keep getting a lot of buzz ... How can I get rid of it? ... Should I undo the truss rod adjustment?
When in doubt, it's always a safe bet to undo whatever you've done and start over on a new path. Truss rods are by definition curved in order to increase or decrease the "bow" warping of necks over time. When you turned your truss rod, you increased the forward bow of your neck (which is why the fret buzz decreased). However, when you lowered your bridge saddles, you lowered the string action thereby increasing the fret buzz. I highly recommend you return your truss rod to its original position and raise the bridge saddles to remove fret noise. As I mentioned above, you may also need to consider tweaking or raising your nut if you intend to stick with 11's.
A little note about string gauges. Thicker / heavier string gauges will yield better tone. It's also better for low tunings like Drop D (or Drop C). It's also a safe workout for your fingers. If you're up for the challenge, consider playing 11's. I know that some players like Steve Vai are famous for super-thin strings (because they do a lot of hammer-ons and pull-offs) but I've also known many metal players with super heavy strings. Totally up to you. Don't let anyone tell you one is better than the other (except tone-wise, this is objectively a fact).