We were in a similar situation (Speediness of offers and trying to time the rental lease & sale) and it's just too much pressure. It also sounds like you've already started to consider walking away.

This is the very last purchase you want to make while having reservations. Daughter loves it? I bet. She'll probably love the next one, too. And the one you tour that has a pool. And the one that you tour that has an extra-large kids bedroom. Kids' opinions change quickly, and if she's unhappy about it it'll be a day-long thing at most, where if you're unhappy with it, it'll be a 10-30+ year ordeal.

I won't say don't do it. There's 999 other things at play here that could make it a great opportunity for you, but so far, you've said:

1) You're possibly rushing into things, and at the very least trying to time things perfectly because of your current situation.

2) Your kids emotions are weighing on you.

3) You're not getting a great deal.

4) The sellers aren't working with you like you'd hope (This is the least of your concerns, IMO).

5) You're being pressured by outside forces beyond your control.

6) The issues presented could be very minor, or very major if initial inspections weren't fully up to par.

Quoting another comment because I think it was worded well:

Now the emotional: The worst thing that happens is you lose this house, plain and simple. You don't get hurt, no one dies, and you move on to a new potential purchase. So on a scale of 1-10 how much does losing THIS house mean to you.

When we went into this situation one thing we were told (Actually by our current landlord, of all people) was that 'We missed out on a house too that we were dead-set on. We thought it was perfect in every way. Life's funny like that, because the house we ended up with was better in every single way for us.'

Just putting all that out there.