I was denied entry into USA due to some CBP notice (most probably a misidentification on the no fly list). I then applied for DHS TRIP and I was cleared for travel to the USA. However, my F-1 visa then went into administrative processing and it hasn't been issued yet. Would I have any problems applying for a Canadian student permit? Also can I apply for both USA and Canada visas simultaneously?
Since both countries / policies are independent. There shouldn't be any problem.
To apply for an F-1 you should have been accepted to an US school. Same goes for the Canadian student permit. So, I find it strange that you want to get accepted to schools in two different countries and that you want to apply for visas to attend both, but technically, that shouldn't be an issue.
Keep in mind that both permits expire soon after your education program is done or you are no longer enrolled.