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I have a 7 year old budgie named L'il Bird who has recently shown some interest or comfort in being more tame. She was originally my sister's and my sister never had the time to properly train her. She is finger trained, but is usually terrified to leave her cage and bites your finger like crazy when you try to take her out of her cage.

Strangely about 2 years ago she started coming out of her cage on her own, just sitting on top and sometimes flying to another part of the room. Then she started coming to the open door for kisses. She will not bite my face at all and will chat away to me, but as soon as she is near my hands/arms she will bite non-stop. I've tried to get her use to coming out of the cage, but she will not stop biting, even once she is safely on my hand she will just keep biting the finger she is sitting on. If she flies off or goes to my shoulder and I go to pick her up she will step onto my finger no problem. I do think once she gets more use to being out she will be a happier budgie, but I'm not sure if I am causing her way to much stress trying to get her out. Do you think I should just let her be and not bother trying to get her out on my finger? Or do you think it would be a good idea to slowly work with her? If so, what should I do to help her trust my hand?

Take her into a small enclosed room, like a bathroom. (make sure you close the lid to the toilet). Very calmly and gently work with the bird by offering her your finger. Press one finger horizontally up against the birds belly back towards the legs. Try to get the bird to step up. If he flutters away pick her up right away and set her back on your finger and try to get her to step up by repeating the process with the other hand. Eventually your bird will get the hang off it. Work with her like this for 15 minutes a day.

You can tame a old budgie if you spend alot of time with him he will get used to you .the best way to train your budgie is when you trim his/her wings so then you can practice taming your budgie so he/she wont fly away

That bird is probably as tame as she should be.
Do you really want to clip the wings of adult birds that might never have
been tame and clipped as babies ?
Hens might be nippy, and that situation might not change.
I'd be happy enough with the birds if they are happy and friendly.