Seems to me that on arrival, your only options for health insurance coverage are travel health insurance, private German health insurance or (if you are from a EU country and until 3 months in Germany) the EHIC health insurance card. Travel health insurance seems the simplest and cheapest for a few months or even a year, but they often exclude people with pre-existing conditions.
I've read that it's difficult to switch from private to public and would like to know if I might face any difficulties.
It is not at all difficult if your eventual job in Germany is as an employee. Because then, there are only two options and both allow you to be part of the public German health insurance without issues:
You earn ≤4950 EUR/month (so-called "Versicherungspflichtgrenze", value for 2018). In that case, you are obliged to be in the public German health insurance, with no option to choose otherwise. So they have to take you.
You earn >4950 EUR/month. In that case, you have the right to voluntarily join the public German health insurance even without having been in there before (source).
I also have some old injuries. Would anyone be able to confirm that the state system accepts non-citizens with pre-existing conditions?
If you're obliged to be in the public system as an employee, they have to accept you (and I've yet to hear of a case where they did not). If you can only voluntarily join the system (for the case of earning >4950 EUR/month), I am not sure.