For single words, I would say tolerate is the closest, that's the word I would use to describe the things you listed. Maybe 'accommodate' as well but that implies more something you are enduring for now but hope to be rid of before too long. 'To bed in' sounds a little awkward and you run the risk of causing offence, as someone might understand it in sexual terms. 'To grow attached to' might work better, implies that two human beings have adjusted to each other so much they are essentially one person.

No, it doesn't! In English, anything involving the word "bed" suggests sex! In fact, without the word "in," "to bed so-and-so" can mean "to have sex with so-and-so." Here's another phrase that does NOT work: "to break in." We speak of "breaking in" a car engine or a new pair of shoes. I can't think of a short phrase for this in English. I would just say "get used to each other" or "get to know each other."

I'm thinking "adjustment period" but not sure if it's what you're looking for. And this is a stage not a process.