You can communicate telepathically with your familiar and perceive through your familiar’s senses as long as you are on the same plane of existence. Additionally, while perceiving through your familiar’s senses, you can also speak through your familiar in your own voice, even if your familiar is normally incapable of speech.

Does this mean I can be inside my genie's vessel and still see through my familiar's eyes

Well - Bottle Respite says that it is an extradimensional space - exactly like a Portable Hole, Handy Haversack, Bag of Holding and Magnificent Mansion - and in most cases that means it's considered to be on another plane. But not always.

Since some extradimensional spaces explicitly say they are on another plane like Portable Hole does - and some don't like Bottled Respite - I think it's safe to say that if it doesn't say it's another plane then it isn't. Maybe it just extends space in the current plane? In that case I think you could see through your familiar's eyes while in the Vessel.

It's a bit wishywashy though. I don't think extradimensional spaces are ever actually defined anywhere.

Basically - don't be surprised if your DM says no.

Might vary by DM, but I would say no. The familiar is still capable of following any orders you left it with though.