#1: Heat death currently seems like the more likely answer - the universe seems to be expanding at a faster rate as time goes on.

#2: Assuming no true source of randomness other than initial circumstances, I fear you'd be right. Even if there are infinite ways that an initial big bang can happen, over enough iterations we would fall into a cycle. One universe would lead to another that eventually leads back to the start of the cycle (simply by virtue of these cycles existing, not being infinitesmally unlikely, and enough iterations happening). However, don't despair. Even if this is true, it wouldn't be you, it'd be a clone. And the clone might take dozens, hundreds, or millions of universes to finally happen again.

#3: Assuming there is a source of true randomness (as there seems to be in the quantum foam), nah. Chaos reigns over such large timescales, so one quantum fluctuation going wrong at t=1 second makes the universe not resemble anything we're used to.