Should the general build additional bases?
No. A single base is the way to go. The general should just save up to 800 gold units and build both bases at once.
How many workers should the general build?
Five workers, to get a total of 10.
When will the general have completed three bases?
295.5 seconds = 4 minutes and 55.5 seconds. This is two minutes after the first mining site is built at 175.5 seconds.
What build order achieves this time?
Build five workers, starting immediately. One will see that there's enough gold to build a worker as soon as the previous one finishes. So, the limitation is the 17-second build time. These workers are then finished at
17, 34, 51, 68, and 85 seconds.
I used a spreadsheet to compute how much gold one has at each of those times by updating the previous gold with the number of workers and mining rate.
Time Workers Minerals Time to 800 Total time
0 5 50 225 345
Worker built 17 6 56.6667 202.8333 322.8333
Worker built 34 7 74.6667 189.4285714 309.4285
Worker built 51 8 104 181.5 301.5
Worker built 68 9 144.6667 177.2222 297.2222
Worker built 85 10 196.6667 175.5 295.5
After 5 workers, just wait until you have 800 minerals to build both mining sites. The "Time to 800" column says the total time its will take to reach 800 gold if you build no further workers, and 5 additional workers optimizes that time. It allows the bases to be started at 175.5 sec and complete at 295.5 sec.
Why produce that number of workers?
A worker pays for itself after 75 seconds of mining, and so after 92 seconds counting building time. Building a sixth worker would not pay for itself as there are only 90.5 seconds remaining at that point. Building fewer workers doesn't help either, as can be seen from the spreadsheet. At the time the last worker is built, there's more than 92 seconds left before building, so it's worth it.
Why produce that number of mining bases?
Any additional base will have no benefit to use. We are nowhere near the cap of 24 workers on a mining base. Such a base would finish after at least 120 seconds, at which point we've already found it's not worth it to build workers, so additional production won't help. Each base it only worth it to reach to end goal. Since building a base occupies the worker that's building it and slows down mining, it's best to wait until reaching 800 minerals and starting to build both bases at once.