How to deal with money and death
In general, your build in early game does not matter much. Just make sure you phaselock those mutants as often as you can. If you are still uncomfortable with your build, try shifting into Harmony to heal better.
As for money, no, you can't store it in the bank, but just stop caring! Every time you die, you lose 7% of your current money—the less you have, the less you should be afraid of dying. I've never had problems with money—even if you miss that rare gun in the shop, it is not like you won't find a better one soon.
There even is a reason and a way to keep you money low voluntarily: Slot machines. There's a low but reasonable chance to win Eridium there, and as cost of playing rises with the level of main quest, there's zero point in saving for later—money inflates, Eridium doesn't! My personal way of playing is to burn all my money in slots after each trip to battlefield. Longest forays usually net me mostly two to three "rounds"—that is, gambling sessions that end in full inventory (of course, buy inventory upgrades first).
Gamble!
- No money - no fear of death!
- Accidental good gun!
- Average of 0.1916 bars of Eridium per pull!
- Eventual "Jackpot!" badass challenge!
The only thing you need a lot of money for is "For the Hoard!" achievement - and accumulating 3,000,000 is fairly easy even at the end of Normal, and trivial at the end of TVHM.
1) Is there a way to store my money in the bank
Unfortunately no. You can store money by trading to another character, but this requires an intermediary player (trade to a coop buddy, trade back to another char you don't use). But I wouldn't consider this worth it unless you're protecting your hoard of $5 million before going off to fight a Raid boss and thus dying literally 20 times.
2) Should I respec? I have 5 x ward, 5 x Kinetic Reflection, 1 x Converge.
Playing better will always help, but there's nothing wrong with that build per-se. If you're dying too often it's more often a case of you being underleveled or out geared. Personally I'd recommend Flicker and the Cataclysm tree, but if you're having trouble killing things, leveling up generally helps more than skills in the first playthrough. Get a couple levels ahead of your enemies and you'll usually be good with only modest skill.
3) Can I just loose my money and get enough later in the game
Basically yes. The price of guns goes up linearly as you go on, you'll be selling guns for $1k or so by the end of the first playthrough (I think, been a while since I was that low level). Money isn't that significant except for buying ammo (almost never a problem) and buying gear at shops (item of the day is usually the only good stuff).