I'm making my own Pad Thai sauce and I'm not sure if I should strain out the tamarind pulp or if it can stay in the sauce. Or is there some type of prepared tamarind which dissolves into the sauce so there's nothing to strain?
It's essential to get all the debris(seeds, large fibrous bits, skins etc) out of the pulp. I normally add a little hot water in a bowl with the block of pulp, be careful not to add too much or it'll end up watery - better to add more as you go, wash your hands thoroughly and break the pulp up by hand. Then strain out the seeds etc and it's ready to use.
You can buy ready to use Tamarind puree in shops but it's normally not a good.
There's more info on working with Tamarind pulp in the footnotes here(as well as a good Pad Thai recipe) http://chezpim.com/cook/pad_thai_for_beginners
The pulp is what you're actually trying to obtain from tamarind. What you should be trying to strain out is any seeds or any chunks too fibrous to be considered pulp. It is sometimes possible to add liquid and smash the tamarind several times to extract more and more (progressively diluted) pulp each time.