Since also off-site solutions are discussed, I'll add one more. StackEdit was mentioned a few times on this meta. It is free, but for some payment you can get additional features. (IIRC export to PDF is one of the options unavailable in the free version.) You can find a brief description of this software in the readme file in their github repository.
- Supports MarkDown + MathJax (i.e., the syntax you are used to).
- Allows to edit existing Markdown files from Google Drive, Dropbox and your local hard drive.
So if you edit your list of questions there, you can have it available in your Google Drive and have easy access to it. I can imagine that making a short list in a format similar to this could be a reasonable solution for your needs. (Just a hypothetical example - I linked to random questions instead, just to have working links. You can basically edit the list any way you want, including some comments etc.)
In fact, I think that having some way of formatting, adding comments, dividing list into sections, etc. makes this actually more useful than just having some list of questions on this site in a format similar to your favorites. (Of course, the favorites have the advantage that they are on-site and thus they are available more quickly.)
My questions about Banach categories
Probably there are also some other solutions where you can use the syntax from Stack Exchange (Markdown+MathJax) and keep the document somewhere online. I will add link to this discussion on meta, which mentions some editors: MathJax: better way to prepare a Math.StackExchange question? And you could also have a look at some of the applications listed in MathJax In Use page on the MathJax website. (StackEdit is metnioned there, too.)
I have just tried mathB.in - however, it has disadvantage that url changes after a new edit. (And I do not have any experience with this page other than the test I did right now.)