I can answer the first one.
If your tractor is used in any form of agriculture, horticulture, livery, pigs or cattle the tractor you are using must comply with the Safety Legislation which includes Noise and Welfare, what ever the age or whoever is using it. That is the Law.
This means it should at least have a safety frame. The frame may be removed for working in low buildings, say, working with a loader loading a spreader. If you the pick up the spreader to go and spread the load, the frame should then be in place.
The tractor should also comply with the noise regulations which specify the noise level reaching the drivers ear and the time he is allowed to operate in that environment without a break. This break could be up to a week after a couple of hours work, in a worst case scenario.
Then there is the welfare legislation that takes into consideration dust levels and seating. The owner of the tractor is held responsible for the Health and Safety of their selves, their employees and any other persons in the vicinity. The Regulations state that an owner can be fined for allowing an infringement, the employee can be fined for the same infringement as they and the owner are jointly responsible. In the case of an owner/driver he/she is fined as both an employer and as an employee.
If a tractor is registered as a historic vehicle and only used on the show ground for display purposes then there is no need for a safety frame.
The grey area as far as I can see is tractors after 1970 that were equipped with a safety frame or cab from new. The Law states these should not be changed or modified yet people remove them because they look better.
What should be remembered is that if you are operating a tractor without a cab or frame and you unfortunately injure yourself it is your responsibility and you may be liable for any injury or damage to third parties. In the case of any accident the noise like a jet engine you may here will be the insurance companies distancing themselves from your claim.
We do not need frames or cabs on our tractors because they are not used for work. (are they)?
I worked for some years explaining HSE Legislation to farmers, food producers, garages and workshops and grain storage companies. I was used as a lead instructor for the local HSE offices and worked with inspectors to get the Regulations over to farm staff. I even instructed the Ministry of Defence site managers using civilian workers on Lakenheath and Mildenhall Air Bases. That was a bit interesting!!! I did Health and Safety evaluations on Ford's Dagenham site where the maintenance staff were driving tractors and trailers inside the factory, beside the production lines, clearing the waste. That again was interesting especially the cardboard compressor which was outside in the rain, with three phase electricity panels open and no protection covering the buzz bars, all surrounded by a deep puddle of water.
Regarding the tyre question a lot will depend on the land and the ballast on the tractor. Heavy land requires a narrower tyre to get the best traction unless you use a wider tyre that is heavily weighted and so cuts in to get a grip.