Why is it oikea puisto and not oikein puisto?
'Oikea' is an adjective and 'oikein' is an adverb. E.g. "It is correct/right." = "Se on oikein."
Why is "on" at the end in some cases but in the order in others? Any way to tell?
"On" often follows the subject, even in questions which is why it often goes to the end in those, unlike English which likes to change the word order with questions.
yes, I am also interested to know if there is an easy rule to remember
what is the difference between "tuossa" and "tuolla"?
'Tuossa' is a bit closer than 'tuolla'. Tuossa = there; tuolla = over there.