Is the driveway existing at is full width or did the application involve the extension or replacement of the driveway. Or, did you extend the driveway without detailing it on the plans?
That would be bizarre if you are replacing the driveway as well. If it just the crossover and outside of the RPA then they may not need to as there would be no impact if the drive surface isn't changed. You may also not need planning if the road is unclassified although this depends on who you ask. I have a colleague that works as an LPA planning manager and he is adamant that you don't need planning for a dropped kerb as it forms part of the highway. His opinion is that what you do need consent for is the bit that connects the crossover to the private property, not the kerb itself, and this is only for classified roads (there is conflicting info about this on other LPA website though). I have seen crossovers done under PD though and subject to planning in different areas. You will always need the consent of highways though, unless its a private road. Are you sure it is planning consent you have and not just permission from highways? I have come across this misinterpretation.
If it has visual amenity they are within their rights to do so, there are other issues to consider though. They should address your reasons for removal in the decision notice, not just say no because of a negative impact on visual amenity.
If they have given consent for a driveway at some point that would be pretty strong mitigation as long as you haven't done further works without consent. Even resurfacing driveways potentially need planning consent unless you discharge the surface water on site. You can replace driveways under PD if you discharge surface water on site but PD doesn't supersede TPOs.
No, it doesn't override the TPO unless you showed the tree to be removed on the plan, you can only do works that are required to implement planning consent. If you have applied to extend the drive and then had that consented (without condition), it would be strong mitigation for any prosecution as they have essentially told you that you can build there. You cant just remove the tree though. If you only have consent for the crossover (outside the RPA) and you have then extended the drive either under PD (e.g. by putting some gravel down, etc.), or without consent by extending it using road stone and tarmac then that could be a contravention. If the drive hasn't been built to spec previously it could also be an issue.
Doubt it. There are ways to engineer driveways with trees in mind. It could be that the drive wasn't built using a tree friendly spec as conditioned by the original consent from the previous owner. That isn't unusual. I have successfully defended appeals via the planning inspectorate for trees within the middle of driveways where the owners are saying they restrict parking and turning space.
There isn't anywhere near enough info here. It will need a site visit and more info on the planning history. Where is the site?