Is the applications fo funding and application for a PhD place treated separately when writing your own proposal? If so how does it work?

Depends really on the institute. When I wrote my proposal (almost 3 years ago now), I applied to a few different universities with it. In two cases, there wasn't really funding or place advertised, I contacted the supervisors casually. On another occasion, there was funding advertised but they were looking for new proposal.

There tends to be in all departments more students then funded places a year.

I mean for ESRC/AHRC etc don't you need an offer of a place?

With ESRC funding it depends on whether you are applying for a quota place or to the open competition.

My understanding is that departments with quota places can give these quota studentships to whichever students they wish. They therefore invite applications, hold interviews and give the quota place studentships to students on the basis of this procedure. Applying for non-ESRC university funded studentships will follow a similar process.

However, when applying for the ESRC open competition you must apply to, and be accepted by a department that is ESRC recognised. The department will then support your application to the ESRC for a studentship in the open competition. It’s up to the ESRC who they give places to through the open competition.

Hope this helps

adding to that... yes, but even if it is a ESRC quota studentship in question, the money and the place to study are technically two different things. the department might decide to give you a place but not the funding. then it would be up to you to either go somewhere else or try to find other funding. still, the place and the money is linked in that you won't get the money without the place
I think it is different with other research councils, but at least in the ESRC the open competition is also bound to departments (that's what we've been told anyway) - it is departments who put forward applications for the open competition and the number of such applications they can put forward is limited, so there's no avoiding the department.