Welcome to the community...And I'm so sorry for your loss...And we all thank you for taking-in your friend's Green Cheek and making him a member of your own family. Parrots are unlike any other type of pet, they are extremely intelligent (that of a 3-4 year-old child), they have excellent memories, they use logic and reasoning skills, and they are extremely emotional, social Flock-Animals who bond very closely with their owners. So while this type of transition and huge life-change is never easy for a parrot or the people who adopt them and take them in to their families and their homes, the fact that Gizmo already knew you well before you brought him into your home and your life is a huge deal, and will no-doubt make this huge life-change much easier for him than it would be if he were to be adopted by a total stranger. So thank you so much for giving him a loving home, the parrot community is much appreciative.
Keep in-mind that Gizmo is going to be out-of-sorts for a while, not only has he suddenly lost his mate forever (and doesn't know why), but he's also in a totally new home/environment. Parrots are extremely territorial, so it's a good thing that you brought all of his things to your house. Keeping him in his own cage with all of his own toys, dishes, etc. will help his transition quite a bit. As far as him "rubbing his beak" on his cage, toys, perches, etc., usually this is a territorial action, like an "ownership" kind of thing. It's also thought that after they dip their beaks into their Preen-Gland and get Preen-Oil on their beaks, they rub it on things or on people who they are claiming as their own, by rubbing their own "odor" (called their "Cologne" by Avian Scientists) onto the possessions and the people who they are trying to either attract, or that they are marking as their own (the Gland at the base of their tail on their lower-back, which excretes "Preen-Oil" that they dip their beaks in and then rub all over their feathers when they preen themselves)... This is called "The Cologne Theory" and is not anything to worry about, he's most-likely rubbing his beak/Preen-Oil on his cage, perches, toys, etc. because they are his territory, and his territory is now in a new home/environment, and he wants to make it known what belongs to him. This is totally normal and not an issue. Just as an FYI (I don't know how much Parrot experience you have) they also rub their beaks on things right after they eat to get the food/water off of their beaks. This is why it's good to always buy at least one Cement-Perch and put it inside of their cage, because it will keep both his toenails and his beak trimmed and prevent the need for you to take him to an Avian Vet just for toenail trims or to have his beak filed with a HAND-FILE (NEVER allow a Vet to use a power rotary-tool on his beak, only hand-files to round the tips if necessary, but the Cement perch addition, if he doesn't already have one, will help cut this down tremendously)...
***Typically when a bird has droppings on the feathers around their Vents, their Tail-Feathers, etc., this is one of the most-definitive signs/symptoms that they are sick, usually with some kind of Gastrointestinal Infection, either Bacterial, Fungal (yeast), or both. I don't know if your friend took Gizmo for a complete Wellness-Exam with either an Avian Specialist Vet or a Certified Avian Vet (CAV) only (no General Vets at all, and no Exotics Vets unless they are also CAV's or Avian Specialists as well), but it's a really good idea for you to start doing so, and now would be a good time to do so, since he does have some droppings sticking around his Vent area. That being said, when you combine the droppings around his Vent area with a broken tail-feather on the underside of his tail with the fact that you just moved him from your friend's house to your house in a vehicle (I'm assuming) and probably while he was still inside of his cage, it's quite possible that he broke the tail-feather during the move and also got the droppings stuck to feathers during the trip as well. *****This of course would not apply if you know for certain that the tail-feather was not broken right after you brought him to your home and must have happened since he's already been inside of your home (since you've had him at your house for a week now)...OR If you have been seeing droppings stuck anywhere around his Vent or his tail-feathers on multiple days or at multiple times and this isn't just a one-time thing (whether just Feces, just the white Urates, or both, doesn't matter)...Again, typically whenever a bird has droppings stuck around their Vent, on their tail-feathers, or anywhere else on their bottom underside, it's an indication that they are sick, and since you just brought him into your home and since he's going through a huge life-transition, and since I'm sure that your friend had not been able to take him for a Wellness-Exam in the recent past, it's a good idea that you locate either a CAV or an Avian Specialist Vet and make an appointment for him ASAP for a complete Wellness-Exam, which should include #1) Complete Fecal-Testing consisting of BOTH a Gram-Stain and Microscopy, which they do on-the-spot in-house using their own Microscope, and also a Culture & Sensitivity, where they send a Fecal-sample out to the lab they use and the culture is allowed to grow-out on a plate over a few days, and you usually get the results back in 3-4 days; and #2) Routine, "Baseline" Blood-Work, in which they do a simple blood-draw from one of the two Jugular Veins in his neck and then run all of the most-common, routine blood-tests that will give them "Baseline" levels that future blood-work can be compared to if he ever becomes sick in the future; this routine, "baseline" blood-work includes all of the basic blood-tests that we as humans have done during a yearly physical, such as a CBC, Nutritional and Metabolic Panels, and very important are both Liver and Kidney Panels, as pet/captive Parrots have a huge tendency to develop Fatty Liver Disease due to inactivity and a diet that is high in Fat and Sugars. There is an excellent, worldwide CAV/Avian Specialist Search-Tool which the link to is all over the forum here, and that lists ONLY CAV's and Avian Specialists and that's it. I see you're in Florida, and we have many members who also live in Florida, so if you let us simply know the town/city in Florida you live in, I'm sure that many members can make CAV/Avian Specialist Vet recommendations to you...Something that you learn very quickly is that a good Avian Vet is hard to find, and unfortunately that most Exotics Vets do more harm than good to Birds and also Reptiles. So it's extremely vital to find an experienced CAV or Avian Specialist Vet, because Avian Medicine is unlike any other, it's extremely specialized, and birds can easily die due to an Exotics Vet who doesn't know what they are doing.
As far as why he's biting at your fingers, I don't think he's actually "biting" you; Green Cheek Conures are notoriously "nippy" to begin with, mine definitely is, but I'm going to assume that he is also very stressed right now, doesn't know what's going on, where his mate is, when she's coming back, why his cage and other territory is in a new house, etc. So I'm not at all surprised that he's acting differently towards you than he did when say you were over at your friend's house playing with him, holding him, etc.
***What room of your home do you have his cage located in? I ask this because even though he knows you very well, he's not "bonded" to you as of now, and you are going to have to approach this entire situation in the same way you would if you brought home a new parrot that you just met. Yes, it's different because you have known him for some time, but the entire situation is now totally different, the dynamic between you two has drastically and dramatically changed, and this is no longer you just playing, holding, petting, visiting your friend's parrot, who he is bonded-closely with. Now YOU are going to be taking the place of your friend, and hopefully Gizmo will first give you his trust (you have to earn his trust first), and after you earn his trust then hopefully he will bond just as closely with you as he was to your friend. So it's very important that you treat this like you're starting-over with him, and that you take things slowly with him... You want to make sure that his cage is located in the "main room" of your home, or the room where you and the others who live with you (his new "flock") spend most of your time whenever you're at home. Being that he's a social Flock-Animal, even if you're not directly paying attention to him he still wants to be around you/near you and the others, so he can see you, and this is going to help to earn his trust and strengthen the bond between him and everyone who lives in his new home. Try to spend as much time as you can with him each day, and try to give him as much out--of-cage time every day as you can. But just having his main-cage in the same room as you whenever you're watching TV, reading, on the computer, gaming, eating meals, etc. will help quite a bit.