Is "có" the general Vietnamese word for "(to) have" (and in extension "(to) exist", similar to Chinese 有)?
Yes, another similarity of (South) East Asian languages. :)
có = 有 = มี = ရှိ = မီး = nga = na = មាន
From left to right it's: Vietnamese (có), Chinese (yǒu), Thai (mii), Burmese (shí), Shan (míí), Jinghpaw (nga), Lhaovo (na), and Khmer (mien).
Does con dơi mean bat together but con mean child on it's own? The drop down under con says child.
That's correct. "Con" is the classifier for animals (to accompany nouns) while stand-alone "con" means child. You will learn more in skill Family below.
Can you say "có một con dơi ở cái nhà ga"? As I saw another sentence using "cái" with "đu quay".
This is a good question and would like to hear the actual answer myself but would also like to speculate that the answer may be no as nhà ga (the train station) is a place where as đu quay (the ferris wheel) is a thing more than a place (it could indicate a place because of its size but it no less a thing), and cái is a classifier for things.
Có is what makes it different. Một con dơi (a bat) có một con dơi (there is a bat)