Hey! Welcome! Let me do my best to answer a few of the questions.
Q1 : In the store there is a chest system.. what exactly is this?
While are few are Obsidian originals, most of the treasure cards come from Skull and Shackles, Wrath, or character decks. They were implemented to add a little flair to the game, and give the players some extra incentives to help out with replayability (get gold, find cool loot, repeat).
Each additional copy of a treasure card you find is like having another physical card in the game box. While treasure card math isn't entirely straightforward, it is generally true that that more of them you possess, the greater likelihood the game will deal them into a location deck. There is a quota involved, so if you are playing with treasure cards on (you can choose to disable them in Settings), the regular Rise of the Runelords cards generally won't be flooded-out by boosters.
When you open a pack of treasure cards, aside from them being added to your account, a courtesy copy is deposited in the "Unclaimed" section of your stash if you want to immediately put it to use. This was done to address a common complaint of "hey, i just opened a gazillion chests, where are all those cards I found??" The treasure cards and stash system can easily be abused to gear-up low-level parties with end-game cards, but however much you choose to "cheat the system" is up to you.
Q2 : There is a section in the store called "enchantments"... what are these thing?
Enhancement cards are single treasure cards you simply buy to add to your collection. They follow the same rules as above, but you are just buying a single card instead of a random pack of whatever. And like the above, you're also getting that free "courtesy copy" each time you purchase one.
Q3 : Then there is "Runes" and "Chams"... what are these
Runes and charms are consumable items Obsidian added to the game. They are entirely unnecessary to complete the game, but can really help out if you get stuck on a scenario. Runes are activated from the overworld map and do some beneficial thing for a certain period of time. Charms are used in-scenario by clicking the backpack icon. Mechanically, Charms are very similar to a side deck of banish after use cards that all party members have at their disposal.
Q4: Is there anything else coins are used for?
Not really, no. You used to be able to buy scenarios or characters with coins, but the in-game economy got a big realignment a year or so ago. Generally speaking, gold buys treasure chests and consumables, but content requires cash.
Q5: Can you increase the size of your "vault" ?
No. The stash is limited to 10 cards, shared across all your parties. It presently cannot be increased. There is no limit on the number of cards in your collection (the game box), or the number of cards in the unclaimed section of the stash.
Cards in your stash can be sold off after each scenario. Doing so does not permanently remove them from your collection (nothing will), you're just selling them back to the "box" for a few extra gold.
As far as other changes, scroll down to the bottom of this post for a list of edits that Obsidian made to cards. Some cards didn't translate well to the digital format, but all of the modifications were made with Paizo's blessing. Unlike tabletop, permadeath is off by default, but it can be reenabled if you really want to. The only other major change that comes to mind is the difficulty system and random modifiers. Playing a scenario on Heroic or Legendary adds one or two random modifiers to bump up the challenge.
Good luck, and feel free to ask if you have any other questions!
Edited by Ethics Gradient, 30 July 2018 - 08:01 PM.