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.