Just a few thoughts for you...
My own experience with the AR-7 (Survival Rifle) hasn't been that positive. Not as reliable as a 10/22, or even a Marlin 60, not as accurate as a less-expensive bolt gun. The "breakdown" feature is novel, but if you're truly in a "survival" situation, you're going to wish for something with more power than a .22 LR.
I like the .223 and Howas/Vanguards, but I'm not sure the Talon isn't a solution looking for the problem? The .223 is forever easy-recoiling, the Talons I see have a "recoil reducing stock" (more $). Oh, and while the .223 is fine for coyotes, it'll destroy a bobcat pelt and it's on the light side for deer. The Talon is definitely radical-looking, but I'm not sure you wouldn't be better off with a standard stock in a Vanguard for about $300 less that you could then apply to optics or upgrade from your next choice...
...The Remington M770. Now, I'm a huge Remington fan, but this rifle (as well as its predecessor the 710) hasn't impressed me. It screams "cheap"! At least the "low-end" Model 788 of years past had a deserved reputation for accuracy and reliability. It's probably going to come with a bottom-end scope that'll get knocked to pieces in short order by .300 Win Mag recoil. I'd look instead at a 700 SPS, or possibly another Vanguard or the Savage/Stevens offerings. You can still find these for under $500. Just me, but the .300 Win Mag is too heavy for deer. I think I'd consider the .30-06. I've killed elk and deer with that caliber for years.
The Mossberg 500 has a great reputation and loyal following. I've found plastic parts where I expected metal in the couple that I've disassembled (maybe cheaper versions than what you're looking at?). 12 ga is good all-around.
9mm is a very forgiving pistol caliber, and it'll still do the job provided you give it quality bullets and put them where they need to go. Heaven help you though if you face an intruder with hardball (FMJ) 9mm ammo.
My wife started out with a Benelli Nova Youth in 20 ga (about $300 if I recall). Some issues with FTFs, but she finally insisted on buying a used 1100 in 12 gauge because the 20 was a handicap shooting trap and wasn't that much less recoil. Some of the 870 Expresses have been noted for rough chambers, resulting in some FTEs. Most important thing is that the shotgun fit her, don't rule out cutting down a full-size stock to get her to the correct length-of-pull.
A .380 is better than a rock, but you might evaluate other alternatives. I would see if she can handle 9mm (my wife does) and if she can, get her pepper spray as well for daily carry. Pepper spray going off in one's purse is inconvenient, but not nearly so much as a .380 going off in one's purse. The .380 will get the job done, but she's going to have to be very close (too close?) and very determined to keep shooting untll the threat is down.
Being on a budget may not be the limitation you might think it could be. There are lots of good firearms available within the limits you specify, and some for considerably less (the Vanguard, for instance). Put good glass and mounts on your centerfire rifles. The "made in China" label is to be avoided at ALL costs - it'll follow you into the deer woods until you replace it with solid, name-brand stuff.