Doing extract, PM, or all-grain is not related to the need for a starter. As for what kind, usually recipes will have it listed. For certain styles it can be pretty important, for others it can really come down to preference. For example, I like to use WLP001 and WLP013 in my house - depends on what flavors I'm looking for (or not looking for).
And sure you can wash dry yeast - after the initial use you would treat it the same as a liquid yeast. Just don't make a starter with dry yeast the first time you use it, it has been packed such that it is not necessary.