Maybe it's a CA thing. Lol I would always ask if kids are invited, and understand that some parties are adults only. For my Sprinkle party coming up, I'm specifically telling people that it's kid friendly and I'd love to see my 2 year old have fun playing with their kids. I know my friends would automatically wonder if kids are ok, so I'm putting it out there that, yes, bring them.
Be prepared. Maybe buy some easy baby proofing items for cabinets or doors (you'll need them eventually anyways). Bubbles makes kids happy and keep them amused. I will say that you may not want a 2 year old coloring unattended. They will most likely draw on the walls or furniture (and floor). Balloons that you blow up, so no helium, and no string are a big hit for young kids. They love to kick and hit them (and it's pretty harmless). And as a rule of thumb, no small toys smaller than a golf ball. If you have a spare tv, it wouldn't hurt to set it up and ask the mom if he has a favorite show or movie.
I wouldn't be ashamed at all to tell your cousin to take the 2 y.o. outside if he's misbehaving while you do a game or something that requires attention on you or someone talking. Two year olds are a handful at times, and loud. But it's pretty normal. Some kids are more bratty or spoiled than others which make it more stressful.
Hope you have a great party.