THIS ANSWER IS CURRENT AS OF PATCH 2.0/REAPER OF SOULS
The answer to this question is now a lot simpler than it was when the Auction House was around, and different difficulties were more ambiguous about their effects. Now you can increase your gold per hour in THREE primary ways:
Bounties and Nephalem rifts
After completing Act V, you will unlock access to Adventure mode, which will allow you to undertake bounties and occasionally Nephalem Rifts. In each of these modes, you are given a specific task to undertake in certain parts of the game, usually killing a certain mob or clearing an area. Once you do that, you are rewarded a substantial sum of XP and gold, in addition to any that you earn along the way by killing mobs or completing a side quest like a cursed chest. In Nephalem rifts, particularly, mobs come fast and furious which create ideal conditions for lots of gold drops.
Raise the difficulty
The higher the difficulty, the more gold will drop. However, it is up to you to find the sweet spot between what difficulty allows you to move quick enough to make the bonus worthwhile. i.e. is it better for you to run 4 bounties per hour on Torment I or 6 per hour on Master?
Normal Enemies have 100% health, do 100% damage. No XP or gold drop
bonuses.
Hard Enemies have 200% health, do 130% damage. XP gains and gold
drops both boosted by 75%.
Expert Enemies have 320% health, do 189% damage. XP gains and gold
drops both boosted by 100%.
Master Enemies have 512% health, do 273% damage. XP gains and gold
drops both boosted by 200%.
Torment I Enemies have 819% health, do 396% damage. XP gains and gold
drops both boosted by 300%.
Torment II Enemies have 1,311% health, do 575% damage. XP gains and
gold drops both boosted by 400%.
Torment III Enemies have 2,097% health, do 833% damage. XP gains and
gold drops both boosted by 550%.
Torment IV Enemies have 3,355% health, do 1,208% damage. XP gains and
gold drops both boosted by 800%.
Torment V Enemies have 5,369% health, do 1,752% damage. XP gains and
gold drops both boosted by 1,150%.
Torment VI Enemies have 8,590% health, do 2,540% damage. XP gains and
gold drops both boosted by 1,600%.
As in vanilla D3, increasing your gold find stats will provide an increase to the amount of gold dropped. You should balance this, however, with finding gear that makes you more effective at killing mobs, thereby increasing your throughput. Luckily, on most gear, gold find is considered a "secondary stat" and does not take away from primary skills like dexterity or crit chance. You'll need to find your own balance of quantity of kills vs. quality of kills.
You can also increase your gold find through the Paragon leveling system. Keep in mind, as above, you'll be taking points from other areas that might make you more effective in combat, such as "life on hit". But this is a good way to increase your gold find without finding specialized gear. Since you can reset paragon points at will, this may be a good way to increase gold find on difficulties where you don't need the other utility stats.
Nobody outside of Blizzard knows the specific number yet, but the number is at least 100,000,000,000 as that's the max amount you can list an item at in the auction house.
As with every game, there is a technical limitation, but that depends on how the number is stored. A 32 bit signed integer would put the cap at roughly 2 billion(or 4 billion if unsigned), but since the max bid amount clearly surpasses that, I suspect that they may be storing it as a 64 bit integer, putting the technical cap at 18,446,744,073,709,551,615.
Of course, Blizzard is free to make it lower than that number, and probably have done so.
You do not put gold in your stash. If you die, you do not drop gold, and gold is shared between all your characters of the same type (hardcore/softcore). You can trade Gold to your friends in trades.
Best Answer
THIS ANSWER IS CURRENT AS OF PATCH 2.0/REAPER OF SOULS
The answer to this question is now a lot simpler than it was when the Auction House was around, and different difficulties were more ambiguous about their effects. Now you can increase your gold per hour in THREE primary ways:
Bounties and Nephalem rifts
After completing Act V, you will unlock access to Adventure mode, which will allow you to undertake bounties and occasionally Nephalem Rifts. In each of these modes, you are given a specific task to undertake in certain parts of the game, usually killing a certain mob or clearing an area. Once you do that, you are rewarded a substantial sum of XP and gold, in addition to any that you earn along the way by killing mobs or completing a side quest like a cursed chest. In Nephalem rifts, particularly, mobs come fast and furious which create ideal conditions for lots of gold drops.
Raise the difficulty
The higher the difficulty, the more gold will drop. However, it is up to you to find the sweet spot between what difficulty allows you to move quick enough to make the bonus worthwhile. i.e. is it better for you to run 4 bounties per hour on Torment I or 6 per hour on Master?
The table above comes from this site.
Increase your gold find percentage
As in vanilla D3, increasing your gold find stats will provide an increase to the amount of gold dropped. You should balance this, however, with finding gear that makes you more effective at killing mobs, thereby increasing your throughput. Luckily, on most gear, gold find is considered a "secondary stat" and does not take away from primary skills like dexterity or crit chance. You'll need to find your own balance of quantity of kills vs. quality of kills.
You can also increase your gold find through the Paragon leveling system. Keep in mind, as above, you'll be taking points from other areas that might make you more effective in combat, such as "life on hit". But this is a good way to increase your gold find without finding specialized gear. Since you can reset paragon points at will, this may be a good way to increase gold find on difficulties where you don't need the other utility stats.