Hmm, this is not a straightforward task at all.
PhotoMechanic does not have any method of determining whether or not files are identical. It does check to see if two files would end up with the same name (e.g. after a rename or when ingesting), and will solve this by appending a letter or digit to the filename. This is not what you are looking for though.
What I can suggest you do, however, is the following:
First open all the folders you want to merge into a single contact sheet. With an empty contact sheet, add all folders to it by e.g., right clicking the folder to add and then choosing "open in current contact sheet" (choose the "and subfolders" variant if you want to add whole trees of folders).
Once you've added all folders you want merged, optionally choose a sorting method (with lots of images, some sorting methods can take a LONG time). The sorting will determine the order in which they get renamed (and thus the order of any sequence number you add to the filename).
Now select all image files (cmd/ctrl-a) and choose file|copy/move photos. In the resulting dialogue, set the correct options (e.g., select "Move photos (delete originals)" if you want to actually move the files to a new location, or leave it unchecked if you want to copy first, leaving the originals intact. Certainly check "Rename copied/moved photos as" (more on this later). Now select the destination either by selecting "Copy/Move to this location" and browsing to the correct folder, or leave it at "Always pick destination" to defer this question to just prior the actual copy/move. In either case make sure you select a folder that is not yet part of the folders you actually try to merge (create a new one and call it "Merged" or somthing, you can always rename/move it later).
About the renaming. There are many possibilities here, but something like "{datesort}_{model}_{seqn}", will be a simple implementation of what you want to achieve. You can set the format of the seqn variable with the button that will popup after adding {seqn} to the rename string.
If and only if all your files are still in their original naming (e.g. DSC_####), you can also use the original frame number in your rename string. In this case use "{datesort}_{model}_{fnum}".
Couple of notes on the renaming used here:
{seqn} will continue to count up, and will not reset when e.g. the date or the camera model changes. Numbering will be following the order of the images in the contact sheet.
{model} generates the camera model with its full name, including the brand name. If you shoot Nikon, you can easily chop off this name by using {model:6} instead. With Canon, this works too, but in this case you might also want to chop of e.g. EOS, so use {model:10} for instance. In a mixed case, this is more tricky (advanced: use code replacements in this case).
There's a lot more possible with this, but above should get you started. Come back to us if you need more info.
Cheers,
Hayo