It's all the same system - you get a set amount of gold at the end of the game, which is an abstract representation of how much gold you'd have if you sold everything and split up the proceeds, plus any actual cash or rewards you might be getting. It's already got your #1 - #3 covered for you.

You can then use that gold to purchase "stuff" - whether it's loot from that game's Chronicle Sheet, loot off one of your older Chronicle Sheets, basic gear from the core books, or equipment from the additional resources list (subject to the rules on that page).

One note: anything you find during a scenario can be used for free within that scenario (including single-use items), but you have to purchase it if you want it for use in future scenarios. You do not have to pay for potions and scrolls found and used during a scenario out of your gold for that scenario. (See p.19 of the Guide.)

Here's how I handled this role

In an older D&D 3.5 edition game I played, we had some similar growing pains. The problem became obvious when we discovered that the random decorations we had taken when we'd nicked everything not bolted down from a mansion were worth a lot of coin. Now this was before the glory of Google Sheets and readily accessible laptops so we opted for a nice graphing paper table. I was not decided for treasurer but I was decided for quarter master. So my advice is as such.

Split the Job

A treasurer is nice and all till you find out they've had the healing potions all along and the Fire resistance potions right after your red dragon fight. So split the load. Either physically have two people or have two separate lists. Equipment and Consumables vs. Loot. I can't accurately recall the exact terms of the table but it was something like...

Item|Date Received|Quantity|Description|Value

Split between two lists, we were able to keep a fairly accurate record of our findings. But it's not useful unless everyone buys in. Which leads me to a second point.

Log Everything and Who Has it

It is inevitable that someone will want something from the loot list or the gear list. They might be decorating their newly bought cottage that totally won't be destroyed next week (probably by them) or they might actually need to have a Potion of Giant Strength to match that inordinately strong arm wrestler. So the job of the treasurer is about logging as much as it is about tracking. It's important to keep track of who owns what item because it could be very important for burial purposes (whether you loot them or not is a matter of contention) or for situations where the party is split. This leads right into the last point.

Log the Total Amount of Money but Each Player Keeps the Coin

No one likes handing their gold over. Even when they understand the reason it is an unfun thing to do. So don't do it. Sure there should be a party fund, a tax on all proceeds for party-centric purchases like a permanent teleportation circle, a castle, etc. It puts a target on the treasurer's back both from bad guys and bad acting players to have the whole coin purse. To me, it's just not worth it. So log incomes and expenses as if all the gold were concentrated but leave each player to their own devices with their money.

Caveats

As with all things, caveats exist. Things that are unlikely to ever be useful again like that dungeon key from 5 months ago can just go under a misc. tab where it is never to be seen again. Same goes for non-value items like letters, body parts, or other accouterments. This is clutter and makes for more work than is necessary.

One thing that really can get you all messed up if you use tables like this: Do Not Erase Anything. Once something's used, either strike through the list, or put an x next to the item. For consumables, just mark an x for every one used.