She doesn't even know that I have intentionally bought spirits online.
All she knows is that I buy pretty rocks and they 'accidentally' have stuff in them and I ask her what the spirits in them are doing.
And F has bitten my hand before, first is when I had her near the cat and she didn't like that and a few more times when I asked mom what F was doing when I was holding out her vessel on my hand.

I agree with L2R here... Don't rely on your mother to tell you anything about your spirit. The spirit is only going to show her what it wants to, and that does not mean it represents the truth at all. You need to work on developing your own relationship with the familiar and practice sensing it with meditation and other exercises, as referenced all over this forum.

There are a couple of things about this entire situation that don't sit well with me. I mean no disrespect about it, but I think you have some misconceptions about spirit keeping.

First, your spirit (the familiar in this case) does not live INSIDE the gemstone. The binding is only a communication link between you and the spirit, in the simplest of terms. So the familiar is not "attached" to the vessel. It can be anywhere it wants to be (in this or the spirit realm for that matter). If it doesn't like your mother, it does not even have to be near her. If it doesn't like your cat, same thing. Your vessel is not required for your familiar to be present anywhere.

Secondly, speaking about most practitioners' bindings, the vessel itself is usually cloaked so no one can "look at your vessel" and see what the spirit is doing. Additionally, like I said above, the spirit is only going to reveal itself to the keeper. It has no reason to trust anyone else so why would it reveal itself to them? You keep asking your mother for verification of the spirit and what it's doing. I would recommend stopping that practice and try to see the spirit in your mind yourself. Chances are you will discover what your mom thinks she sees and what your spirit is really doing are two different things.

Lastly, protection. Spirits cannot usually easily physically interact with people. This often takes energy. Maybe she can bite you because you are sensitive enough to the spirit realm to feel it. I doubt, however, that it will bite anyone else in the physical realm, and certainly not as part of "guarding". It might be able to give them a bad feeling, bad energy, or try to shield like someone else mentioned before or even try to put warning thoughts into their mind, but a physical bite is highly unlikely. I think in that regard you are over-expecting.

One last thing, a familiar is not actually an animal. It might manifest as a bird, but it's not a bird. It won't necessarily behave like a bird either, and that includes biting or being afraid of a cat. I highly doubt it actually cares about your cat being around because it wouldn't be able to harm it anyways. That's not my point, though. My point is that a familiar is a guiding spirit. In many circles they are considered ascended. They were never animals in this world. They are living spiritual entities from the earthen realm (really between the earthen and spirit realm, but that gets a bit off into the weeds), and CH conjures the spirits of those who are dead.

The point I am getting at is this: I recommend that you treat her as a spiritual guide and not a bird. It sounds like you need to work more with her one on one and without someone else's external input, especially your adversarial mother. I also recommend that you work on your telepathy and stop relying on the cards. They actually do not sound very accurate to me.

Again, I don't mean any disrespect in any of the above. I hope that some of this helps.

Judging by the fact that you wrote a few decent paragraphs about F's behavior, I guess that means I should clear things up.
I already know that spirits don't live in their vessels but F likes to stay near or on her vessel.
The three angel spirits and 1 angel entity I have all appear in the hallway without their vessels.
Same thing with the Dark phoenix I have;she doesn't need her vessel since she appears in the hallway.
The wraith I have stays in my room because their vessel is also located in my room. They haven't appeared anywhere else in the house.
As for the Volkh, they don't really manifest unless the cat starts playing with the vessel on the ground, then the volkh appears as a green pink light and chases her around the house.

F seems to stay with her vessel because she can only be seen in the area her vessel is in
(Example: The familiar has been witnessed sitting in my coffee mug when her vessel is on the table near said mug but hasn't appeared anywhere in the house.)

Now, as for the ''Your familiar's vessel is cloaked because you shouldn't be able to see her'' thing, it isn't really true. She either sits on or near my vessel and has been accurately seen as a secretary bird while the other spirits can't really be seen unless they appear in the hallway.

As for the points about ''It's not a bird, it shouldn't be afraid of a cat'' or ''It doesn't have to be in the same area as your mother or the cat since it can teleport back to its realm'' part, F can be seen sitting on her vessel. Sure, she can technically teleport to her realm but I still felt her peck me after I had her vessel near the cat and then I picked it up. I felt F peck me when I had mom look at the vessel in my hand.
Bird or not, F sure bites like one. I didn't hallucinate any of her pecks, they don't feel good.

I'm sorry for this in advance but I'm forced to agree with Reg here. To me it sounds like you're treating an entity like a house pet or a guard dog, and relying on your mother's advice as a way to comprehend the Entity. How much of this is true is a debate I don't want to even address as only you know your journey. I'm just stating what it sounds like when a closer look is taken.

That said, if none of the above is accurate for you at least take their advice when it comes to your mother. She doesn't sound reliable in the slightest- not as a housemate nor as a spiritual companion advisor. Hell, she doesn't even know what you're actually buying, so how on earth could she properly analyze it without being fooled by the Spirit in question? A spirit who does not wish to be seen - won't be seen. I've known many who play with illusions and will often even use those illusions on their Keeper as a way of either playing with them or admonishing them for a behavior. Though that gets into personality and personal Keeper Spirit dynamics.

What I'm trying to say is, if none of the above hits home, dropping ALL preconceived notions and starting fresh without relying so heavily on outside sources (Your cards, your mother, the being's vessel, etc) would be best. For yourself and the being.