Fitotrack. It's like strava, but open source. I use it regularly, and I'm sure it has all you need.

OSMAnd on fdroid can record all sorts of trip data and can spit it out as gpx files, which are just XML and probably pretty parsable in something like Python. That might get you at least some of what you want.

The first thing that came to mind is Strava. It's been a long time since I used it, but I found this thing about Strava Beacon which sounds like it might do what you're looking for.

https://support.strava.com/hc/en-us/articles/224357527-Strava-Beacon

You can use GPSLogger to record it in local or send it to whatever service you want.
If you're into selfhosting you can use traccar which is focused into fleet management so it's easy to get reports on the trips made.

As for your second point, I wouldn't trust the GPS for this, it can say you weren't moving since it only checks every so often to record the data, or maybe it says you actually were speeding because the two points it used to calculate the data weren't the actual points you were at that time.
A dashcam would be better suited for this. I'm not sure how they work, but most probably they can be connected to read data from your car which would be more trust worthy to whoever might decide if you were actually speeding.

Phone track is on fdroid and uses owntracks format. I use it with a custom api, but it is build for nextckoud.

I know magic earth is like waze but has a built in Webcam mode that could record your drive, not sure if you can then export the GPS and speed data though

I have always like GPS essentials for various other location tasks. While I have never used it, there's a track mode under streams. Data can be exported in a handful of formats.