Yes, you can visit the temple mount. Like other reviewers have mentioned, it is currently open to non-Muslim visitors beginning at 7am. Check with the guards the day before you plan to visit to confirm.
The entrance is up a ramp and you will go through security first. They may ask your nationality. (I did not need to show a passport but it might not be bad idea to bring it).
Once you go up the ramp, you may be asked to put on a scarf if you are not dressed modestly. You do not need to cover your hair, but as at other religious sites, women should cover their knees and elbows.
Yes, you can walk around. I went just after 7am and it was very quiet and peaceful. Take a look, take some pictures. Admire the history and be respectful that it is a religious site.
When leaving, the exit is different from the entrance. You cannot return to the Temple Mount through the exit which is another spot in the old city. There will be security there and the street will be blocked off so anyone who you want to meet after your visit would have to wait outside of the barriers.