By time limit, do you mean the Ref? If so, there's a cut off announced and things must be published by that date to be included - published after, and they are eligible for inclusion in the next one. Just doing the research doesn't meet the requirement - it also has to be available. It all goes on a central database at the university, that each member of staff updates for themselves, and then is winnowed out from there. If you have more than four publications, you are usually asked to rank them.
In order to "do" research, you have usually to go through various internal hoops eg ethical approval, but in Humanities and Social Sciences, no-one is usually telling you what to do. If you've won external funding, obviously you will need to spend it roughly as you've said in the application, and follow their reporting requirements etc
Your department might have the odd meeting where everyone updates on what they are doing, but usually, it's something you'd discuss yearly in your appraisal - just future plans etc
You can basically do what you like, within reason - so theoretically, although I'm a development economist specialising in the Middle East and Africa, I could decide that my next research project is going to examine the cost-effectiveness of Obamacare. Except, for various reasons, it wouldn't make sense such as, unless I was spending time in the states for other reasons: so it would be expensive without funding, and there are other people better qualified to work on that, so I am less likely to be funded etc
On a day to day basis, though, no one cares what you are doing, and rarely asks.
I wouldn't get too hung up on the research questions at the application stage tbh: they'll want to know if you have anything yet that's ref-able, and whether you're likely to have by the deadline if not. And it's probably a standard application form/ job spec anyway - so check out your potential colleagues to see how much is applicable in your department