2 situations.

1. An opponent takes control of your char with Melisandre's Favor. This char has Doran's Favor attached. Will Doran's Favor be discarded because of its "Attach to a character you control" text?

2. An opponent takes control of your char with Melisandre's Favor. Then you attach Snakeskin Veil to this char. Will the char return under your control?

1) Yes.

2) Yes.

1) Yes.

Is "Attach to a character you control" a continious effect? It looks like passive?

"Attach to a character you control" is a constantly checked effect. This situation is addressed directly in the FAQ, §3.22:

(3.22) Attachment Restrictions
Any attachment that has a restriction (such
as "Lord or Lady only" or "attach to an
opponent's character") is immediately
discarded from play at any time that restriction
is not met, regardless of immunity. Such
restrictions are constant effects, and the
attachment should be immediately discarded
any time a restriction is not met including
during setup.
For example: Tom has played the card
Motley (Core Set B67) on Kris's House Tully
Recruiter (Core Set S21). Motley has the text:
"Attach to an opponent's character." Later in
the game, Tom uses Seductive Promise (Core
Set T175) to take control of the House Tully
Recruiter. Because the Recruiter is no longer
"an opponent's character" from the perspective
of Tom's Motley card, Motley is immediately
discarded from play.
Unless specifically stated otherwise on the
card, attachments always attach to a character.