Translation:I like swimming.
Think of English words like archery (arrow shooting), equestrianism (horseback riding), gymnastics, etc. Japanese (and also Korean) use Chinese pronunciations in much the same way as we use Latin and Greek for nouns in a specific, formal context.
Oyogi is a noun. Suien and Oyogi mean the same. Oyogu is a verb. This small change can be seen in several verbs or adjectives such as 歩く(verb, walk) 歩き (noun, walking) 遊ぶ（あそぶ）(verb, play) 遊び (noun, playing). It can be too complicated but you will find them in Japanese.
There is also 遊泳 (ゆうえい) in Japanese, though it's not that frequently used, I guess
You are right. That's sounds more natural. But it does work with oyogi as well. "I like to play" will be遊ぶのが好き, but you can also say 遊びが好き which is "I like the fun part (factor) " in designing, for example.
Do i like swimming, or do i like the swimming (as an event) are they distinguishible?
I'd probably never say just "the swimming"; it would have to be followed by another noun like "event" or something.
You can specify that it is the even by saying, 水泳イベントが好きです or a bunch of other ways otherwise swimming includes all forms, but will be implied to olympic if that is the main topic.
Could be translated as love, but there are actual words in Japanese that means love (particularly 愛[する]).
好きcan be translated both ways. Depending on the context, it can be love our like. Some people avoid direct speech so they say " find of" or "fancy". Sometimes using the lighter words or negative of the opposite expression, you can empathize the meaning. Like "we are not amused."
You can either say, "oyogu no ga" or "oyogi ga". But "oyogu no ga" sounds better.
If you want to distinguish this from other sports events which you might not like( to watch) you are better off using the definite article and respond "I like THE swimming". Suiei here implies the event does it not??