Irregular time-series analysis

I'm trying to find patterns in a game event that happens randomly over time. The data looks like this:

ind  start-time        end-time          starting-score     closing-score
1    2020-01-02-13:00  2020-01-02-13:50  10                 17
2    2020-01-04-07:00  2020-01-04-09:30  17                 15
3    2020-01-07-23:00  2020-01-08-01:10  15                 34
...

As you can see, the event (playing the game) happens randomly, its duration is random, and results in changing of the score.

Now, the answers I'm looking for to do my analysis include:

• What is the best time of the day to play the game?
• How long should you play to get the best results?
• Are there any patterns? e.g. if you earn points twice in a row are you likely to lose points in the following round?
• Is it better to play the game everyday? what is the optimal rest time between the sessions, etc.

There are so many interesting questions, but I'm still not sure how to target the problem.

I thought maybe a feature-based analysis could be a good start, i.e. binning the start/end times into different parts f the day (e.g. morning, evening), then I will have a categorical feature, same can be done for the length of the game (e.g. short, long). The first 2 questions might probably be answered using a regression (or logistic regression) model on the outcome points, but I guess I need some sort of time-series specific analysis to be able to provide better insight and I'm not sure what methods I need.

I'm looking for a solution in Python and I have seen packages like traces. I guess the main problem I have is that most of libraries are designed for regular time-series and those that support irregular ones are focused on how to deal with the missing data, while in this problem, nothing is missing, it's just the nature of the data that is irregular.