I played my single-player in fear of not having enough money for this achievement, hoarding my capital and not buying houses when it would have been easier.
#1: The $10k goal is cumulative. You need not have it all at once.
#2: It appears that you don't lose money when you die. (If you do, it was never noticeable for me. This isn't Borderlands.)
#3: All in all, this will most likely come naturally on your way to 100%. I wouldn't go out of your way to get this achievement first.
Tips for making money:
* Go on a treasure hunt. Search "RDR treasure" in YouTube, and finish the treasure challenge, selling the gold bars for at least $1.5k.
* Most bounties go for $200+, especially when returned alive.
* Skin everything you kill and sell it in a far away land. Horses and dogs aren't too great, but grizzly bears skins sold in Mexico works well. Cougars are okay. In the hunting ambient challenge, the final special hunts will bring in big bucks. [pun not intended]
* Gamble. Poker took too long for me, but the Liar's Dice game in Thieves Landing works well. Play with one man at the table, ante in the required $200 and win over and over for $200 bucks a shot.
* Loot everyone. Perhaps this isn't totally necessary, but it was probably my favorite part of this game. $5 here, $3 there, it adds up! (Also, buying the Rabbit's Foot will increase your looting returns by 20%.)