It would be cool to have more than one LPP. Two or more LPPs would be great so that I can breed them.
Would be cool to be able to beam to other people so they can use them to breed with. There could also be genetics/character traits that can be passed down to the offspring. (Like The Sims 2)

they could play with each other!

It would be undeniably cool.

they could have fights and battle for money and stuff

Yes, playing with each other would be cool and adorable. But (and with all due respect to the one who suggested it), please, no fighting puppies!

Are they puppies or kittens?

There was kind of a funny discussion of that very thing here.

Personally, I liked the Tamagotchi type thing, where how you take care of them and play with them is what affects their eventual morphing.

Yes, - good question. Well, I believe they are puppies because I choose to see puppies, - since I am partial to dogs. I also refer to my puppy as "he." Oh, the psychological damage I could be doing to him!