I'm going to go against the majority opinion you've been given and suggest the 20 gauge simply because of the criteria you've set. A 12 gauge can be hard for a woman to handle.
And since you're looking for home use I would not recommend the weapon you have pictured.
For tactical purposes (doors, hallways, stairs, etc.) a pistol grip, with or without a folding stock is, IMO, a better way to go.
As far as the neighbors and penetrations goes...how is your house built? All wood? Brick? Combo of both? Distance between them? That matters when considering ammo. The more durable the construction the greater the choice. (buckshot and bb's OK)
Pistols? Hard to give "good" advice. Anything from a .22 to a .45. A lot is dependent on the user. A small woman might not be able to handle a larger caliber so a smaller one is more practical. You get the idea. Find what is comfortable. The rounds you use can help make up the difference.
JMHO
Name of the shotgun and link to purchase it?
Wood frame & stucco. About 10-12' between houses here.
I would like to buy one of these too. Linky?