Yes, do rinse it. It's always a good idea to rinse any grains or beans (yeah, I know it isn't REALLY a grain), but if you don't rinse quinoa well you'll get a soapy taste.
The texture totally depends on how you cook it...cook it the way you like it. I've eaten it mushy, tender, and sort of crunchewy, depending on how long the person making it cooked it. I don't think any way is right or wrong, it just depends on what you personally like. I like it a bit more tender, but not mushy. I tend to eat it dry though, not with sauce, so that may make a difference.