As long as potential illegal work is an issue, nothing else matters.
You appear to be intending to abuse the privilege of visiting. Your planned activities do not appear to require the time in the US that you plan to spend. If you intend to do a PhD in the US, why do you need to travel so extensively at this time? It certainly appears that you intend to continue your previously stated planned of working remotely.
I repeat that you should engage and attorney and be completely honest with him/her for the best advice.

