Two simple silly questions, just to be sure.

If I control Alfyn Crowkiller that says:

Any phase : "choose and discard a Wildling character from play (cannot be saved) to choose and kill a Night's Watch character with STR 2 or lower"

The wildling targeted must be a character controlled by me, since it is a cost. The cost here is identified by the keyword "to". The Night's Watch instead can be controlled by any player. Right?

Instead, if I control Fallen Brother that says:

"After you play Fallen Brother from you hand, kneel 1 influence or choose and discard from play a Night's Watch or Wildling character "

I can only kneel my influence, since kneeling influence is always a cost, but I can discard a character controlled by any opponent. Right?

Correct on both counts. Costs must be paid with cards you control. Effects can be applied anywhere, unless a more limited scope is defined as part of the effect.