is my thread where I started off talking about the original (red) Sizzix that I own, and ended up talking more about the electronic versions, so that might answer some of your questions. I haven't honestly researched Pazzles much, and it looks like there are some great resources for that!
As far as I know, the Gazelle will do both edge and 'embossing folder' style embossing. Like any of these electronic die cutters, there is less pressure than in a manual machine, so it might not be as deep/clear of an impression. Again, I don't have one so I am going on what others have said.
As far as cutting out stamped images, you can do what is called print and cut with digital images, where you print out the image with 'registration marks' that the die cutter reads and cuts around. The Cameo is supposed to be great and easy for this, but the Gazelle just upgraded their system, so I have heard it is good too. My plan is to make files by scanning my physical stamps into the software and drawing an outline around them (with the software). You would probably ideally die cut and then stamp, but with the Gazelle at least you can choose your origination point. So you put your paper in and you put your blade exactly where you want it to start, making this kind of cutting /seem/ easy.
There are several videos - search YouTube for "Gazelle die cut" or something similar. You can also look for the BossKut forum or the Make The Cut forum for actual users - which I obviously am not yet.
I feel silly trying to answer questions when I don't actually have the machine - maybe others who have one can do a better job!