Does it make a difference whether I use an HDMI to DP or a DP to HDMI cable? Does the conversion work both ways? I'm currently trying to connect a monitor with a DP connection to my laptop with an HDMI connection (I have a DP to HDMI cable).
When in doubt: no. Why else should you specify a "direction"?
In general, if a feature is not explicitly specified, the probability that the product will also master it is low. In any case, you can't complain about a feature that has not been guaranteed.
Don't think so.
Displayport can send an HDMI signal, called dual mode, but HDMI can't send a Displayport signal.
HDMI to Displayport needs an active converter, so to HDMI is usually possible simply passively with an adapter for a few euro
Active converter is relatively expensive, mostly
If your case has Hdmi output from the laptop and DP input from the monitor, I would assume that only 60 Hz or no G Sync works. The two cables themselves still have versions like Hdmi 2.1 or something. If then buy a good cable