List of open problems: pros and cons of making it public
Point to keep in mind:

Asking questions is easy.  Asking questions other people think is interesting is less easy (see some questions on this site).  Asking questions that lead to important results is hard.

Fundamental ideas come up over and over again, so even if they don't take off at first, they'll spread eventually.


Pros:

If some of the questions are indeed interesting, you could help people find good problems.  If you're lucky, you may get asked to collaborate, but I wouldn't count on it.

Such a list, may help get your name out there.  Maybe the questions are great and people think you're brilliant.


Cons:

The most obvious and serious con for you is that someone else solves what you're working on before you do, and a lot of your work becomes unpublishable, and you don't get a job you want.

Some of your questions are trivial/well known and people think you're stupid/ignorant.

Worst case scenario (exaggerated): everybody in the field finds your questions super-interesting and all work on the same question but nobody solves it for the rest of their life and the field dies.


Suggestions
Think very carefully about which of your ideas you want to share, with whom, and with what purpose.  I'm not going to advertise to the world what new idea I'm working on, or even a good one I have on the back burner, until I make serious progress or have compelling evidence.  Because (1) I don't like to compete with people, and (2) if nothing works out (most common) then it's likely either not a good idea or not the right time for this idea.  This doesn't mean I won't tell people, or ask other people related questions, I just don't post a pdf on the internet.  I tell some people I trust, and sometimes this leads to collaborations.  Occasional &quot;further problems&quot; to projects I've done, I mention in papers, but this is when either I don't want to do it myself or I deem it unlikely anyone else would solve it before me.
Also, at least for me, often it's years between first asking a question and answering it, but if I had advertised it, maybe someone else would've done it in the meantime robbing me of the satisfaction of answering it myself.
If you decide you are willing to share some of your ideas on the internet, and you think they may be of wide interest, rather than just posting a list, consider making a blog of some of your thoughts.  If you get readers, this could generate interest in things you like, which might make your work more popular.  At the same time, you can pose your questions in a very informal way, so they're not presented as if they're supposed to be a grand list of the most fundamental problems in the field.
Not everyone has the same philosophy as me, but I would consider being too open about your ideas somewhat risky at your stage.
Oh, and maybe ask your advisor first.