I think "выиграть and проиграть" are kinda hard to be explained by roots. Knowing that высоко- (high in value, status) and выше- (above, fore-) both have the meaning of "up-high", it's reasonable to assume the prefix "вы-" may carry a similar sense. So вы-играть basically means "play up high".
As for " про-", here is a quote from Wiktionary: "(used with verbs) (describing an action that describes losses, expenses, wasting, or forgetting)."
Hope this helps : )
The stress falls on the first syllable. If you're interested, you can find the stress of all conjugations here. https://en.wiktionary.org/wiki/%D0%B2%D1%8B%D0%B8%D0%B3%D1%80%D0%B0%D1%82%D1%8C
Yes, выиграть is indeed perfective, and the imperfective form is выигрывать.
(Wiktionary is a great resource for checking Russian verbs... :-) )