I will throw in that if your other half will in theory be allowed to apply for work in the UK without having to talk to the home office (unlike you). Generally the primary person's visa (yours) says something along the lines of no access to public funds (you can't go on the dole) and all work changes must be approved by the home office. The rest of the family's visas only say no recourse to public funds. So in theory, spouses can apply for jobs, but to be honest, it's fairly hard to land employment with a visa tied to someone else as the applicant can't guarantee they will be around for the long haul. So perhaps take this time to decide to have a free child (go socialized medicine!) and convince the spousal person that they want to be the primary care giver ;) I kid, well mostly… They don't do dual hires by and large, and actual unemployment in the UK is quite a bit higher than the US so when considering any offer, make sure it's enough to take care of your small family on. (Oh, and note that any salary over ~ 80k USD will be taxed in the US as well as in the UK.)
Also, no, there aren't the usual illegal questions for them to ask.