I can't think of a specific rule, but I couldn't remember an exception, so I would say yes

Yes, піаніно is a smaller piano, whereas grand piano is called рояль. There is another word - фортепіано, - which is a more generic term for any type of piano, both small and grand.

You might find these Wikipedia pages helpful:

https://uk.wikipedia.org/wiki/%D0%9F%D1%96%D0%B0%D0%BD%D1%96%D0%BD%D0%BE

https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D1%8F%D0%BB%D1%8C

https://uk.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D1%82%D0%B5%D0%BF%D1%96%D0%B0%D0%BD%D0%BE

But it's worth pointing out that this particular sentence - Я люблю грати на піаніно, - means that you like to play piano in general. Any kind of piano, not just a small one. Unless they are professional musicians, people normally don't differentiate between a small or a grand piano. Even though it's not technically correct, to regular folks any kind of piano is just піаніно :)

Піано doesn't exist in Ukrainian as the name of a musical instrument. It is only used as a term to describe how one is supposed to play a certain place of the musical piece, along with other such terms like pianissimo, forte, fortissimo, lento, allegro etc.

We use піаніно or фортепіано for any kind of piano. Піаніно is more colloquial, I would say.

I know nothing about this instruments, but according to Wikipedia, piano (фортепіано) is the collective term for all instruments of this kind. Grand piano (рояль) is a configuration with the horizontal frame and strings. Upright piano (піаніно) is a configuration with the vertical frame and strings.

I have extended the alternative translations to accept "upright piano," but I'm not sure if this is the most widespread term.