Hi
We picked up our 10 week old rescue lurcher pup last Friday (pics on profile).
He's very underweight, so currently on between 6 and 8 meals a day, depending how many we can squeeze in (he has a knack of sleeping through at least one mealtime a day). The inevitable result is that he is pretty much either eating or pooing at the moment and his poos are not at all nice, so I feel your pain.
The first day we brought him home we put down puppy pads, as his foster mum had used them and he managed all but one 'toilet' on them. We laid one in front of his crate (in case he couldn't hold it when first waking) and one by the doorway of the room his crate is in, plus one just inside the back door.
Second day we removed the ones near his crate and just left the ones by the doorways and since then he's just had the one by the back door.
We've found the best thing to do is spend as much time outside with him as possible, making sure we have a clicker and lots of treats with us and make a huge fuss of him when he happens to 'go' out there. Seems to be working, because other than when dh arrived home from work this evening and pup got over-excited and forgot himself, we haven't had any indoor toiletting at all today, either on pads or the floor.
We are keeping the back door open at the moment, but make sure that someone is always with him and that we take him straight outside if he so much as sniffs the floor, as well as after naps, food, drinks of water and 'mad half hour' play sessions. It's time consuming, I'm getting very little else done and frankly, I'm completely knackered, but I think it's well worth the investment in time and energy, as every positive reinforcement he gets for going outside, drastically increases the likelihood that he'll repeat it and quickly learn that going inside is totally non-rewarding compared to going outside.
As for the crate. Our pup is in his overnight, but still gets up at 2/2.30 ish and again at 6.00 ish for a wee and needs some reassurance around 3 and 4 o'clock. I am currently sleeping on the sofa so that he doesn't disturb the rest of the family and we have gradually moved his crate from alongside the sofa next to me, to the other side of the room to the kitchen since last Friday, to get him gradually used to being alone. He now wakes and whinges a bit about 3 and 4, but goes back to sleep if I shush him and say 'settle'.
During the day I leave the door open and he goes in and out of it whenever he likes, plus I pop him in there if he starts to look sleepy and close the door while he's asleep so I can get some jobs done (like feeding the dcs! ). I also pop him in there if I can't watch him, eg when I go to the bathroom and when the dcs are eating, because, being a lurcher, he can already stretch his paws up to the edge of the table and is verrry interested in their dinner. He definitely sees it as his safe place, as he takes himself in there if he is tired or a bit unsure of something, like when a helicopter flew over the garden yesterday.
Finally, it's very early days. Going to a new home is very stressful and can affect some pups much more than others. Often this shows itself up through upset tums etc. Also, the food he's on might be having an impact. If it's one that has a lot of fillers etc in there will be more waste, than if he was on a natural diet or high-end, grain free kibble. Depending what food he's on, you might want to think about slowly transitioning him onto a food that will result in less waste products.