Sure, you can do both.
I suggest you download the manual (in the downloads section) and read it, it's well worth it. Especially the sections on variables, which are very powerful in Photo Mechanic.
In PM you can have up to three info labels below each contact sheet thumbnail. Using PM's number of variables, these can show whatever information you want, of course including the date:
Just go to preferences, contact sheet, and there set Labels: to Filename + 1 label.
In the field below, you can now enter the text including variables you want to show. Enter {date} (including the quotes). You can click variables... button to see a list of all posible variables and their explanation, there are also different formats of the dates and times you might want to use.
For renaming, same variables can be used, either manually, or even during Ingest, to rename your photos for the filename to include date and time taken in some nice format for sorting plus the original filename's photo number.