are these shoes a true size 9?

I have no idea about a size 9 shoe, but I wear a size 6 and my character shoes are a perfect fit. I would say that the sizes are pretty accurate. The shoes is very comfortable and hard wearing, there should be elastic by the buckle for ease and comfort and if there is a difference in size it is probably in the width. I bought these shoes because they are wide fitting enough for my arthritic feet. I hope that helps.

Do they have a suede sole?

No - they have leather soles. Lovely to dance in and very comfortable.

are these shoes suitable for spinning when jiving?

They don't have suede soles so not as smooth on twists and turns as conventional ballroom/Latin practice shoes.

if my nike sports trainers are a size 9 what size should I get of these?

im a size 5 but when i oredered the 5 they were a little tight i sent them back and ordered a 5.5 theses were much better. i would definately try the half size bigger