Our protagonist, Euclid, has devised a scheme to beat the house in Math World Las Vegas: 

He bets a dollar, double-or-nothing with even odds, on red in roulette. If he wins, great, he's done. If he loses, he doubles his previous bet and repeats until he wins, which with probability 1 he eventually will.

, even though each bet has zero expected value, Euclid is certain to make money with this strategy. But what about his 

: He either makes $1 off the bat or he loses some number of times first. If he loses $1, loses $2, loses $4, then finally wins $8 -- lo and behold, he's netted $1. That's true for any possible losing streak (except an infinite losing streak, which has zero probability), so he always makes $1.

: Euclid is making a sequence of bets, each of which has zero expected value. The sum of any number of zeros is zero. (Also he has an infinitesimal chance of losing infinite money so that has to factor in.)

1. How much money does Euclid start with?

Infinite money! Or, rather, we'll say he has unlimited credit. Any finite amount of money he wants to bet, he borrows it as needed. This is magical math world.

None! Again: math world. Euclid wins each spin with exactly 50% probability.

3. What if the expected value (EV) is infinitesimally positive?

I think the EV is either zero or one but just in case, we'll count infinitesimal as zero. So that would be a YES here.

4. What if the EV is undefined or the answer is "it depends"?

We'll add clarifications until we get a meaningful EV and thus a definitive YES or NO.

5. What happens if Euclid has an infinite string of losses?

He gets a -$∞ payout! Being math world, all the infinitely many roulette spins happen instantaneously.

Consensus in the comments. Failing that, expert consensus. Bet with caution until we've hashed out in the comments what that should mean! And 

be sure to ask about anything missing from the FAQ before betting

[Ignore any auto-generated clarifications below this line. Clarifications aren't official until added to the FAQ.]