1. On the map, we’d like to filter by post type, to either show all or one type of posts.
This is not possible within Toolset generally.
Toolset Views, the tool with which you will build your Custom Search (Filter) does not allow front end filters by native post data like it's author, type and so on.
These features are planned, but I cannot state any ETA on them.
A possible solution to this issue is to store the Post Type information in a taxonomy or field.
Then, you can either manually or with custom code, update that Post Field or taxonomy each time you create (or update, for older posts) a Post.
That Field or taxonomy could later be used to create a Custom Search on the Front end allowing the same as filtering directly by post type.
Any Custom Search that you can add to a Post View you can use as well on a Map.
What is to consider, maybe it's not even required to have 3 different Post Types.
Try to keep the number of Post Types at a minima, differentiate them with Taxonomies, then use Fields to add information that you can not add to Post Bodies, fore example, because you need to enter it in a different value for each post.
2. In addition, with the event-types, we’d like to be able to sort and filter by a date field attached to the post types.
This is entirely possible with a custom date field.
Later, you can use that Field to build a Custom Search about dates.
But, this will be either by a fixed date or a date range (between 2 dates).
No Search can be built (on the front end) by single months or years or days).
That would, as well, require a "mimicking" type.
This means, you would need to have a taxonomy or Field that stores the information of "Month" or "Year" as a single value.
Then, that could be used as a Front End Search.
Now, in the backend this is a bit different, here you can build a Query Filter that allows you to filter as you wish it, but that will not be editable by the end-user of your website, it is a Filter of the query, which filters our the posts before they even reach the end-user on the front end.
3. After these post filters have been applied, is it possible to search for all posts within a geographic region such as a province or a state?
No, Toolset Maps allows a search by distance or location but not by state or city.
I mean, you can of course search by +3000 KM from Paris.
That is the same as searching "in Europe", but it is not the same as searching explicitly by Country alone.
Toolset Maps all in all will only help you to "plot" the results a Map and it can do this well, the problem of this project is more the Query that you need, and the Front End Filters.
I suggest to use less Post Types, more Taxonomies and Fields, and then this search will be possible to be created and plot it's results on the map.
These documents will help you during the process:
https://toolset.com/documentation/user-guides/front-page-filters/
https://toolset.com/documentation/user-guides/#filtering-views
This may interest you too, as it has similar aspects of your goal:
https://toolset.com/2015/04/how-to-create-a-news-site-using-wordpress-if-you-are-not-a-php-developer/
