I don't think they do get to spend gold before Gold Digger, actually. The mini-campaign rules specifically say to perform a "Travel step" before Gold Digger, but they don't mention performing a complete Campaign Phase (which includes shopping) until after that quest.
The rules do say that the heroes save any "gold and unspent experience" until later, so yes they would get to keep this money for future investment.
Yes. Agreed. I also think so.
But they do get to spend xp on skill and OL do get to spend xp on cards.
Setup: Perform all normal rules for game setup. In addition, each hero
receives 4 experience points worth of Class cards and 100 gold. The
overlord then receives 4 experience points worth of Overlord cards.
Any gold and unspent experience points may be saved to spend in
future Campaign phases.