About Chronomaly Cabrera Trebuchet's effect + pay cost for no effect
Q1:
No you can't. You can only pay for costs with your own cards (so cards you control, in your grave/deck/hand...).
There are some cards that allow this, but it will be stated explicitly on the card.
E.g. the kaiju monster or 'chimeratech fortress dragon'.

Q2:
Won't work in this specific scenario for above reason, but to answer in general, this is also not allowed.
In the case of 'Chronomaly Cabrera Trebuchet' the reason is that as part of the cost, you need to target a monster on your opponents side of the field, so if there are none you cannot activate the effect.
To give a bit more obvious example of this, you can't activate 'Monster Reborn' if there's no monster in either players graveyard.