Diagnostic Review: Sunroof and Trunk

Just wanting to check if there's anything else I should do to confirm diagnosis of these two issues.
My battery recently died and after replacing it now I've got these to deal with.

1. Trunk
Usually would not open with keyfob, always opened with button.
Now, won't even open with the manual key.
I'm assuming this is the hydraulic pump, but how do I verify?

2. Sunroof
Left popped up while battery was dead for 2 weeks.
Now, when I try to open with the switch it will only go a couple inches at a time--I can't hold the button down, I have to click it repeatedly. I can hold it to close the sunroof, but now once it's closed if I keep holding the switch to close it will pop the back of the sunroof up as if I'm pushing the switch up instead of forward.
Sounds like a relay to me, but could it be the switch itself? How do I check?