Oh minximoo of course it won't be anything personal, just over worked but the LA could at least ring you and let you know what is happening (or not) SO it is their loss really. I think you are looking at this way too much from the point of will we be suitable - this is a 2 way street you know, and as much as LAs or IFAs assessing you it also means that you should be assessing them!
I think the fact that you posted the stuff to the IFA on Friday a'noon and got a reply on Sat shows that they are very very keen to approve you and sell you to a LA - sorry but it still sticks in my throat about IFAs and the profits they make.
You will I'm sure be on a course before very long and approved. The checks they are talking about are the statutory checks that have to be made - a CRB check (police) definitely. Different LAs and IFAs will have different methods of doing the checks - some want health checks done right away and checks on probation, obviously SS records in your area, and maybe any LA where you have lived since you were 18. This is because if you change your address you will need to be checked out with any LA of that address to see if you were "known" to them for anything child related.
I think the best thing is to talk to the IFA and ask them about the checks - you see, they should have explained them but assume prospective fcs know what they are! LAs are guilty of this too. Ask as many questions as you like and don't be fobbed off when you ask about how many children they have placed for the LA (in your age range) in say 2 years - they might try to wriggle out of this, but be resolute and say you are concerned as to how many times you might be used by the LA. No need to say "how many children have you bought from the LA!" but that is what it means. Ask for evidence of this - no need to be confrontational - just "It would be interesting to see the figures and the areas where the children came from"
I forgot to say that many cash strapped LAs bring children back "in house" at the earliest opportunity because of the costs. You could say that you have heard this and again "how often does that happen" - don't accept "oh not often" ask for figures etc. Ask if they can link you up with an experienced carer in your area to have a chat, which could be very revealing.
Incidentally Looked Aftered Children rarely come in ones, would you be able to take a sibling group of 2 if necessary.
Above all please please stop being so subservient YOU are like gold dust to LAs (wouldn't think so of course) and to IFAs.