Yeah, in the store you can get a few LP from watching videos.
Check the cloud (There's a menu button somewhere, then hit the blue cloud) to see if you might have a more recent save there.
Out of the two options you've asked, the mansion is much better.
Yeah, I think those kinds of things are EA's LEAST priority in this game.
Don't Type Like This, for starters. My cat once dug up 1 LP. They rarely dig up LP, but it's possible.
No, they're completely unrelated. To grow a toddler you need to bake a birthday cake on a stove (costs 5 LP.) then click on the infant and select have birthday.