How do I unlock the new "Second Wave" options?

Question

Second Wave is now a real thing, released as free DLC.

However, I've noticed that not all of the options are available at the beginning. What are all the options available in Second Wave, and how do I unlock them?

Answer 1

UFOPaedia has a pretty good write up. I'll try to simplify their list, however.

Initially Unlocked

• Damage Roulette - Weapons have a much wider range of damage
• New Economy - The funding offered by individual council members is randomized.
• Not Created Equally - Rookies will have random starting stats.
• Hidden Potential - As a soldier is promoted, their stats will increase randomly.

Unlocked after beating the game on Normal or higher

• Red Fog - Any wounds taken in combat will degrade a soldier's stats for that mission.
• Absolutely Critical - A flanking shot will guarantee a critical hit.
• The Greater Good - The secret of psionics can only be learned by interrogating a psionic alien.
• Marathon - The game takes considerably longer to complete.

Unlocked after beating the game on Classic or higher

• Results Driven - A country will offer less funding as its panic increases.
• High Stakes - The rewards granted for stopping alien abductions are randomized.
• Diminishing Returns - The cost of satellites increases with every one that is built.
• More Than Human - The psionic gift is extremely rare.

Unlocked after beating the game on Impossible

• Total Loss - Lose all gear from soldiers who die in combat.
• War Wariness - Overall funding levels drop inexorably with every month that passes.
• E-115 - Elerium stores will degrade over time.
• Alternate Sources - The power requirements of all facilities are increased.