What HF bands, times of day and/or conditions might be best for attempting to make non-local contacts using QRP transmitter power (around 1 Watt) and a small indoor folded or loop antenna?
The lower the frequency, the larger the antenna that is required to be resonant, so if you're aiming for something small then 10m might be best. That band is usually best during the day, and during the summer.
If you want to go with a slightly longer antenna, 15m is also good, either day or night.
As for the antenna being indoors, that could work, but usually (especially with QRP), you'll find that you get the best results with HF with the antenna outside.
There is no particular band more suited to QRP operation than another. A weak signal is still weak, whatever the frequency.
It's easier to build efficient antennas on higher frequencies, since the wavelength is smaller, and antenna designs scale with wavelength. While it's always possible to electrically lengthen an antenna with loading coils, capacitive hats, etc, this makes the antenna less efficient, making poorer use of your already limited transmitter power. So, you likely want to stick to frequencies high enough where you have room for the antenna.
Being indoors is never good: it means you can't get the antenna far away from noise sources and you can't get it very high. That's always bad regardless of frequency.
Non-local contacts are possible on any of the HF bands. Which one is best depends on many factors, a large one being luck. HF propagation is significantly affected by solar activity, which follows some pattern regarding an 11 year sunspot cycle, and the obvious day/night changes, but also is highly variable based on just chance. Higher frequency bands like 10m are mostly good during the day. Lower frequency bands like 40m and down are mostly good during the night. The bands between, 30m, 20m, 15m, are somewhere between. Exceptions to these "rules" are common, based on random solar weather. It also depends on where you are trying to communicate: each frequency refracts in the atmosphere at a different angle, so a particular frequency might happen to not work because the signal skips over the intended target. The sun is moving, and so is the ionosphere, so a frequency that's good now might not be good in 5 minutes. That's just luck.
Given all this, it should be pretty clear: we can't predict what the best band will be. You can't get outdoors, you don't have space for a good antenna, and you have limited transmit power. On balance you have a poor station, and there's no way around it. Your size limitations restrict you to the higher frequencies, but on those for which you have enough room, the best band will be determined by luck.