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Teleport is not stopped by normal damage or Soft CC (slows). Hard CC will. Here is the page where you can find details on Teleport. https://leagueoflegends.fandom.com/wiki/Teleport
Now personally, I recommend using Flash over TP when your new. You can try some off meta builds and do well, but you need to have a good grasp on all champions kits to know interactions.
For the abilitys your asking about Urgot R will not end TP unless your HP is low enough he can reactivate it. The first casts is fine. The second is going to stop the ability. Also as far as the second cast goes. If he is outside of Kindred R you die. If he is inside kindred R you live. Kayne Does not have hard CC unless he goes Rhasst. (red). In that form he can stop teleport. However in all 3 forms of kayne if he casts his R on you at the correct time he will travel with the Teleport.
Everything else your asking about will cancel TP.
When your very new. I recommend you try all free week heroes VS AI bots for a game or two. That way you learn what they do and don't have to find out the hard way by playing with or VS them. Reading is fine, but hands on experience tends to teach lessons faster.
The game has a lot of champions in it to help teach mechanics to new players.
Macro is important. Vision is HELLA important. A lot of players treat Mobas like FPS and go after kills. In my opinion The only thing you should care about is whos nexus goes BOOM! And what you can do to make that happen.
If skill shots are giving you a hard time. Try spending a few weeks playing Battlerite on steam. That game is free and has almost no macro so you can get a good feel for Micro skill shot gameplay.