Hey thesko-
This is a pretty wide topic, good questions. Some answers and opinions:
1. The batteries we sell as well as most similar lithium batteries have a protection circuit that cuts off power around 2.8-3.0 volts, the minimum voltage for these batteries. So there is not a safety concern and should only be a minimal affect on battery life by running it 'all' the way down to the cutoff.
2. TinyScreen+ will generally run down to 2.6 volts, but may act a bit differently when the input voltage is below the 3.3V onboard regulator voltage(such as analog reads, depending on reference, and OLED display).
3. 3.5V that you mention is definitely a low state of charge, it could be a good place to indicate a very low battery and prepare for shutoff.
Thanks,
Ben