Setting Rankings for Fixed Time Periods

Right now the way I see it is I have to create a new match type for each month and then a new ranking for each month. Then set up a new page using the Ranking Shortcode (what affect does the date option have on this?) which I can change at the end of each month or change the ranking in the widget.

Creating a match type per month is not needed per se, but it will help with grouping the matches in a view and to select the correct matches in a ranking.

The date in a ranking shortcode is used to show a historical ranking up and until that date. E.g. when your writing a mid month update and want to show the ranking at that point. When using the date parameter you will make sure that the ranking shown in the post, is not changed some weeks later when new results are added.
For your case with the monthly rankings, the date is not used. But you’ll have to group the matches for a given month in one ranking.

As I add new games each week and add them to the current month match type, will this then update the monthly rankings after I enter the scores manually?

Unfortunately, newly added games are not automatically added to the ranking. I only store the match ID’s, not the match type ID in a ranking. So you will also have to also add the new games to the ranking itself.
When this is done, after doing a calculation, the scores are added to the correct ranking. If you forget to add the games to the ranking before adding the results, you can always add them later and do a calculation to fix the ranking.

Finally is this the best way to do this or am I running around the houses unnecessarily?

This is the only way to do it if you want separate rankings for every month and not cumulative rankings, and you want to keep the data for a complete season.
One thing to bear in mind, though, is that the plugin will become slower in the calculation process with every ranking you add. Especially when there is a large user set and a lot of matches. I don’t have any data on it, so I don’t know at what point the calculation takes too long. But might be something you may want to test upfront to avoid disappointments later on in the season.

Another approach can be to just delete the historical data (matches and rankings) from the plugin once a part of the season is complete (e.g. a quarter or mid season). And to just copy&paste the rankings as text in a post if you want to keep those.

Hope this helps.