1) Which would be the easiest language to learn for a Swede with English as a second language?
Latin, not because it is easy, but because learning Ancient Greek is incredibly hard.
2) Which language do you think I will have more use of?
Neither would be anywhere near as useful as Chinese, Arabic, Sign Language, Spanish, et, al, but if you learned Greek you would be able to read the best book ever written (the Iliad,) the second best book ever written (the Odyssey,) the most important book ever written (the Greek New Testament) and the second most important book ever written (the LXX.)
3) If you learn any Latin, is it possible to read anything from antiquity to the renaissance or is there a learning threshold to each of the phases?
There are many texts in Latin, even more in Greek, that are so hard that while it is not impossible to read everything across periods in these languages, it is virtually impossible. Easy texts can be read from any period, but the problem is that there are not that many easy texts. There are more easy texts, I think, in Latin than there are in Greek.
4) If I were to try to learn both, which should I start with?
It does not really matter which you start with. Starting a language is never the problem. The problem is sticking with it. The vast majority of people who start learning either of these languages never learn them. I would start with Greek, though.
Would it be confusing to learn both at the same time?
No, I don't believe it would. The problem will not be confusion between the languages, but the difficulty of each individually.
Let us know what you decide, and keep us informed of your progress.