Can a Lithium Polymer battery charging circuit on Arduino boards also charge Lithium Lion batteries?
There are Lead based, Nickel-based and Lithium-based batteries (there are actually more, but these ones are the most popular). Each need a different charger, but since lithium polymer and lithium ion are in the same group, it should work but I don't have the option to test it out for you.

Q1: Not every battery has a built in over- and undercharge function. Some do, some don't. To find out if yours does, if the internet can't help, tear it apart (be extremely careful about it).

Q2: Depends what you mean by " built in battery charging functionality". If that includes the over- and undercharge functionality, then it will work (tested).

Looking at the board (https://wiki.wemos.cc/products:d32:d32#technical_specs) it seems doable. 

The most important thing is to check, if the supported voltage of the li ion battery equals the supported voltage of the li polymer battery that can be charged with the li polymer charger. Since it says 3.7V on the data sheet, don't get a li ion battery with 3.7V+