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Лапочка is OK but лапа is no good.

As for cute I'd say sth like: милашка, красавчик, симпатичный.

or you may use simply Он такой муси-пуси! :lol:

"Муси муси пуси пуси миленький мой
Я...

Yes, it sounds effeminate. Only girls and poofs would say "какая лапа". Mind you, boys will say какая лапа in the sense 'that's some hell of a paw, that!'. As for other words that only boys or girls...

In the plural it's лапочки (perhaps also лапоньки):

these puppies are so cute = эти щенята такие лапочки

but boys don't use such language.

girls would probably say "какая лапа". OMG, I've just realised, it's so ambiguous - какая лапа could mean "he/she/it is so cute" (not only about a puppy, BTW) but it could also mean "well, that's one...