Putting him out is not the same as going out with him. He may well not be toiletting if you just put him out the door, as he probably wants to get back in.
You need to take him to the spot you want him to use, wait quietly until he goes, then praise / reward him. This means every time, not just when it is fine, daylight, convenient. This will give him a positive association with being outside and a positive association with toileting outside.
Be patient. When I got my last puppy, it was a little local drought breaker, and it rained all day for days. So I got wet, he got wet, the toys and treats got wet, but he learnt to toilet outside, quickly, on command, ever time I ask him to.
No, I stay with him, sometimes for 30 mins or so.
He likes playing outside during the day and sometimes does not seem interested in going to the toilet untill we get indoors!!
I suppose he will learn.