That is a dilema @PiotrL, especially involving family.
If the firearm is used in an illegal manner by the family member, will I be held liable as the firearm is in my name?
I think the best way to look at this is to compare it to a car. If you loan your car to someone, you are generally not liable for the illegal or negligent acts of another person with some exceptions. For instance, if you knew they had a history of bad behavior and you still let them use it. In Kentucky, criminally this could equate to complicity and civilly it is negligent entrustment.
The same should apply to a firearm.
But, if you are in a state that requires reporting that a firearm was stolen, this could possible bite you in the butt.
If I request again for the firearm to be returned, and they don’t do it, are there any legal options I can follow to recover the firearm?
Yes there are legal options, both civil and criminal. If the gun was loaned in your home state, everything would happen in your home state. If in the relatives state, then action is taken in their state.
Criminally, failure to return something that belongs to someone else is, in Kentucky terms, Theft by unlawful taking or disposition, specifically in your case it would be disposition. Again using the car analogy, if you rent a car and never return it it is a theft by unlawful disposition.
In a civil suit you would usually seek damages for the actual cash value of the firearm and any modifications. However, if it is a special weapon, collectors piece or family heirloom, then you may be able to get equitable relief to get that gun back.
Not related to this situation, but related, if I sell a firearm to a private party, and they use it illegally, will I be held liable?
You need to follow the law of your state when you sell a firearm. Some states require all sales or transfers to go through an FFL while many do not. Kentucky does not. That being said, you cannot KNOWINGLY transfer a firearm to someone who is prohibited under federal or state law in Kentucky. This includes handing it to a person at a range to try it out. Knowingly is generally looked upon as she knew or should have known that this person was a felon. , although it does not require an extensive inquiry. As others have said, there is value in running a sale through a FFL so that you are relatively certain the sale is not an issue. This also puts that person in the system with that specific firearm.
I do not know what your state’s “registration” requirements are. In Kentucky, there are none, although if the sale goes through an FFL, it will show up in NICS.
I sold a firearm in 1997 in KY that I purchased in 1984 in PA. About 10 years later (2007ish), I got a notice from federal court in Louisville advising that they had a weapon that I once owned and that I could get it back after it was confiscated from a thief. I let the AUSA know that I had sold the firearm and nothing else happened.
Hope this helps.