One screw is quite an adventure. The lock is a Yale HE. The thing held by one screw is called the curb. On this thing is the wheel post with three wheels. To change the combo this this has to be removed. And when it is missing the lock can be opened without a combo ... if there were not the relocker to the right of the lock. If the screw falls out and the curb wiggles its way out the arm on the right side will move and block the bolt work. Safe permanently closed.
As it is necessary to change the combo the curb is meant to be removed. So I don't see an issue removing it, but be careful with closing the door. I think you can remove the door cover too, to get access to the complete bolt work. You can clean and grease it. Sticky grease is not so bad there as you have the handle to apply force. Everything behind the the curb should not be greased, but only slightly oiled and probably only on the post the wheels are sitting on and perhaps some steel parts to prevent rust. To much oil or esp. grease can lead to wheels and parts stick together, so that one wheel drags the next one and the safe doesn't open. When you have removed the door cover you will also understand the relocker and then know how to handle it.
I would check three things.
1) To much sticky grease?
2) With the curb installed dial the combo, remove the curb and then look how the gates (the slots in the wheels) are positioned. Are they well aligned or is something of?
3) After removal of the curb you can see the drive cam, which is connected via a threaded spindle with the dial. Cam and spindle are locked together with a spline key. Is this all put well together or is the spline key loose?
The wheels on the curb should be so called mesh change wheels. The changing procedure is like described in this video:
https://www.youtube.com/watch?v=x6L_jQJRQXQ If you disassemble it, it is generally a good idea to make photos or notes of the order of parts, because things can be slightly different from the video, e.g. how the wheel order is marked.
I would look for a complete set of screws to prevent an accident. I would also disassemble the wheel pack as described in the video and wipe of every part. This is often enough, but you can put parts into an ultrasonic cleaner or clean it with break cleaner (everything there should be metal unlike in the video). Before reassembly oil or grease the post very slightly, i.e. put the stuff on the post and wipe it of again. I don't have any experience with graphite and can't say a thing about it. In the video you see bent bronze disks at the bottom of the stack. If your lock has them you can use them to control how much force is required to turn the wheels. If it is to little wheels can continue to turn on their own when they have momentum and then it is important to dial slowly to get an opening. I think it is good if you can feel the pickup of wheels. I would keep the combo as it is for the moment and not separate wheels in inner and outer ring before the old combo works reliably. The dial could also be removed for some lubing, but I don't recommend it unless it is necessary. To remove the dial the spline key has to be pulled and later reinserted. Every reinsertion of a spline key it sits less and less tight in the slot and can then create problems when it gets loose. If you do remove the dial, then at least do not remove the dial ring and (if it can be removed) the lock case. The realignment can be a pain.
Good luck.
PS: 1882 is the patent date. It only means that parts are not made before this date, but they can have been made decades after.
PPS: The safe is - besides the screws - in a really great condition as far as I can see from the photos. Very often they are heavily worn on the outside. Also often a few or perhaps all of the acorns decoration the hinges are missing. Please cherish it!
PPPS: If you hold up the relocker you can retract and extend the bolts while the curb is not installed you can see how the mechanism interacts with the lock.
