I do think uneasiness is always worth taking some notice of. I personally wouldn't read too much into the 5 weeks to get back to you thing, am not sure is an indication of anything - often seems to happen - companies are desperate for someone and then take ages to decide.

About taking time off I would be direct and ask about the deal with Fridays - it could just be they don't want someone taking Friday as holidays - but are quite accepting about time off for sick children. The answer you get about the Friday and how they answer might give you more of an indication about an attitude to needing to take days off for sick children in general. You already feel uneasy about it, not much to lose by asking the question, if they suddenly took back the offer then you were right to be uneasy about taking the job.

Good luck for other one today