When you email a professor to inquire about an available PhD position is it a good idea to ask if they have any available projects?
You should do basic research on the topics the professors is researching himself. Afterwards, you may either see whether there is a project around where you could supply your workpower, or you have an idea on a concrete project where he could extend research. (Keep in mind, he might need to get a funding for that).
If you do have an own project with independent funding, it might be easier - however, it should fit in the reseach area of the professor.
If you just come around, most professors I know simply will skip your application, as the normally have sufficient requests and work to do.
You should first check your program if it is your POI who recruits you or the grad school that does the job.
If it is the grad school that selects the student, keep in mind that whatever POI replies do not necessarily reflect your future. Most of POIs are busy, ie, they have thousands of emails to reply each day. So, this is unnecessary. Worse come to worse, they can put your email address in spam. So, my suggestion tends toward not making any contact with him.
On the contrary, if it is your POI that recruits his students, it is good to observe his research interest from the recent papers. Then, if any of those matches yours, you should mention them in your email. Even though you have some of them in mind, you should also ask POI about his particular research interest and funding. Expression of curiosity and eagerness to work with him is a good sign.