Can you simply crate him, door closed, to prevent him from going into rooms he is not supposed to? He should be perfectly fine in there for an hour and a half.
It's pretty normal for dogs to try to get closer to people, which is why he is jumping the gate to go upstairs. I understand he is a foster, so you might not want to be facilitating that type of bond. He doesn't understand your routine, so it's also normal for him to act a bit restless.
If you don't want to crate him, you can also double up on baby gates stacked one atop the other, so it's too high to the dog to jump.
It's pretty normal for dogs to try to get closer to people, which is why he is jumping the gate to go upstairs. I understand he is a foster, so you might not want to be facilitating that type of bond. He doesn't understand your routine, so it's also normal for him to act a bit restless.
If you don't want to crate him, you can also double up on baby gates stacked one atop the other, so it's too high to the dog to jump.