First off, Welcome to the site DSan.
I have some trigger time on the Remington 870, and while it's not my first choice for a shotgun, it's a very good, solid and reliable weapon. A lot of the 870's are used as platforms for police unit builds and for defense guns. It's easy to find parts for it and everyone can work on them because they are so common.
Here's the problem, if it's for home defense, it's a little big to wield effectively in tight spaces, like a hallway, in a standard configuration. I believe you can order from Remington and get the Knoxx adjustable stock system, and while it's not the best out there, it would help with the overall length, plus it's adjustable to length of pull for you or your woman.
There is nothing wrong with the Mossberg, they make some great weapons, but I would stay away from a pistol grip, no stock, configuration. That thing is going to jump something fierce when you touch it off, and if you don't know what you are doing, you could end up doing more harm than good.
You might take a look at their Persuader Line
if you like the Mossberg stuff. Same basic model as the Remington, you get the 18" barrel and a synthetic stock to help manage recoil.
Now, as for a 12 or a 20. Your size, and that of your woman, should definitely come into play in your selection. A 12 guage is a big kicking round when touched off and you aren't ready for it. If either of you are smallish of frame, I would recommend the 20 as it's a bit more tame. If the recoil isn't an issue, the 12 guage would be fine, but if you had to hit someone with the 20, it's not like they are going to shrug it off. The 20 has plenty of stopping power as well.
All of that isn't going to do you any good however if you don't get out and practice with it. That is going to be your biggest asset because the thing isn't going to do all the work for you. The spread pattern of shot in the 20 to 30 feet between you and the bad guy isn't going to be wide, it's going to be about the size of your fist. So, you can definitely still miss the target quite easily if you don't know what the weapon can do.
As for a round - #2 or #4 Buck is always a good choice as stated above. I like the #2 personally.
Probably the best advice I could give you would be to get to a range with someone that has one of the weapons and try it for yourself.