Is there a reason you don't just get the bike in the eBay link you provided?
You said it is similar to your last bike that performed well for you, and is well within your budget (i.e. may leave money for better locks), so it seems like an obvious choice.

You can find cheaper used bikes, but to go below this price usually takes time, and some ability to fix the problems with inexpensive used bikes. (Good bikes without problems usually cost more.)

An additional 16 miles twice per month shouldn't strain any bike; your last bike would have been fine (if it hadn't been stolen). Many commuters do more than 64 miles per week, so I think any decent bike should work fine for you.

Single speed
Telling us gear 5 or 6 is hard to use, since this will depend on the number of teeth on the front (chainwheel) and rear (sprockets) gears, and on the wheel size. Changing the gear on a single speed isn't particularly difficult or expensive (put a larger or smaller gear on the rear hub), but it sounds like you might need a few tools and some instructions