In general, I would suggest not using the Downtime and Kingdom Building rules, since they are generally written very poorly and have a lot of weird edge cases and odd inconsistencies. That said, it sounds like you are having trouble with the dichotomy of capital earned versus capital spent, and also how capital checks work.
The fact that you are in a Small Town only restricts the number of capital you can spend per day. You can earn as much capital as you want each day, with no real limit. If your capital checks say that you can make 500 Magic in a day, then the fact that you are in a town with less than 500 people doesn't stop you. However, it will take over a month to actually spend all of that capital. The line to be aware of is:
The numbers in Table: Settlement Spending Limits represent the limit of how much Goods, Influence, and Labor you can utilize in settlement each day.
Emphasis mine. The Settlement Spending Limits only apply to the amount of capital you can spend, not earn.
In addition, your capital check values are a little off. I'm going to assume that the bonuses for each building that you've calculated are correct, at least for the moment. For each building, you only make one capital check, probably based on the highest bonus that you have for that building, unless you have a reason to want a different kind of capital. Assuming you're going for the highest total capital amounts, your "take 10" values for each building would be:
- Monument: Influence 16
- Tavern: Influence 37
- Hospital: Influence 61
- Castle: Influence 178
This totals 292, for 29 Influence per day. You don't get to make a separate check for each type of capital each day, you only get to pick one at a time. The line to be aware of is from the Income section:
Attempt a capital check (see the Earnings section) for each building you control in the settlement that generates income and is able to provide you benefits.
Emphasis mine. Since a capital check only lets you generate one kind of capital at a time, your buildings can't make multiple kinds of capital at once.
About weekends: The whole 'weekend' thing only applies to Craft and Profession checks made weekly. Honestly, I'm pretty sure that they just put that section in so that they could divide things by 10 without running afoul of the existing craft rules. In real medieval life, there were no such things as weekends. The idea of taking two days off every seven is relatively new, and is sort of a weird thing to add to a system like this. If your GM wants to restrict work to the modern work week, then that's their prerogative, but there isn't anything in the rules that actually says that you can't work or spend capital on the weekend. If your GM doesn't enforce the weekend, you can spend a total of 5475 capital per year in a Small Town. If they do, you can spend 3900, as you note.
This expenditure can be on anything you have the capital for, but must be done in 15-capital increments. If you want to build a structure that costs more than 15 capital, you spend 15 capital each day until you have spent enough capital to build the whole structure. Note this line from the Construct Buildings action:
How much capital you can spend per day is limited by the size of the settlement you're in (see Spending Limits). Once you've spent the total capital and time needed to finish your building, it's complete and you can use it immediately.
You are correct that earning capital only allows you the opportunity to buy that capital. The Skilled Work rules note:
You must pay the Earned Cost to buy this capital, although if you can't afford to buy all of it or don't need more than a certain amount, you can choose to earn less capital than your check indicates.
There isn't a separate section for capital earned using buildings, but I'm pretty sure that buildings are supposed to use the Skilled Work rules. Like you say, if you produce 3,900 Goods, you need to spend 39,000 gp to actually get those goods. Remember, though, that you can produce gp instead of capital, if you want. There's a different bonus for earning gp, but it works pretty much the same way as a normal capital roll: take your results, divide by 10, and that's what you earn in gp.