Glad to hear Freddie is doing well.
So about the head scratches: I wouldn't worry to much how to initiate it. I would wait for him to show you that he wants one, by rubbing against your finger or something like that. To get him more used to your fingers, I would look for moments when he's otherwise involved, like eating millet out of your hand, and maybe move your finger close to his head so you can give him a scratch if he wants it. I think it's very possible he'll get more used to your fingers
I think that kind of biting is like we'd consider roughousing. Mostly it's unintentional, unless he's mad at you for whatever reason in which case it probably isn't
Either way, as he approaches his "teen years", he'll start to get a little more bitey, but most budgies grow out of it. The way I dealt with my hen's nippiness during her first few months was wait until she bit me, and then put her back in the cage after saying "No, Mallorn." Then I walk away. After a few minutes, you can come back and act like nothing happened, as parrots understand easily the implications and soon won't behave in a way to make you leave the room and stop playing.
A new environment is always a difficult time, but it's a necessary time. Is it possible to get your Nan over to your house before you leave and introduce them in a familiar environment? If you do this, he'll be less apprehensive of spending ten days with them. Mallorn met her caretakers before I dropped her off for the same amount of time and she did really well. Reassure him you love him and you'll be back, and I'm sure when you return he'll be very happy to see you
Birds have a good memory so it's very unlikely that when you come back, he won't recognize things. It may take him a few days to settle in but he'll be back to normal after then
Assuming you got him as a baby, three months is not very long at all. And apart from that, it is possible to teach an old budgie new tricks
Many budgies surprise their owners with a random word far after their owners attempted to teach them to talk. If he's mimicking and interacting with you actively when you talk to him, he may be working on a word already! Use similar words and phrases (Hi Freddie! Pretty bird!) when you talk to him and he may pick one up.
Regarding a second budgie, if you and your husband work a lot, a second budgie boy may be a good idea. However, it's not entirely necessary if he seems active, playful and bonded to you both. Regardless, going on holiday without them while they are in a cold house is not very recommended, as budgies are sensitive to changes in temperature. You could buy a smaller cage (still big enough, but not as big as their home one) to keep them at the home of whoever takes care of them to limit the inconvenience of having to tow around a huge flight cage.
How cute that he loves his toy! It's normal for budgies to become obsessed with a toy. I think that if it's not to serious an issue, you can play with him and his toy during out of cage time. However, if it seems he's really bonded to it and becomes sad when you put it away, I would remove it. To distract him, you can get him new toys and keep him entertained.
I hope this answers all your questions! Keep us updated and I can't wait to see some new pictures of him