They're all fine. "feels like" is informal and more likely to be used in everyday speech. "Seems" in this context is more formal, especially without "like." If you used "seems like" instead of "seems", it would be more informal, but not as informal as "feels like" Using "I feel that" instead of "I feel like" makes the statement more formal, more certain, and more potentially insulting. To seem is more definite than to feel, so in general "seem" is more formal, certain, and potentially insulting than "feel."

All are acceptable, however "I feel *that* you're lying" and "It seems *like* you're lying" is better. "It feels like you're lying" is also fine :)

Yes but because this could be insulting you need to be careful about when you say it.