1. Generally one would cap the termini of the protein if there is a non-native residue at the terminus; the capping neutralizes what would be a non-native charge. If you have the entire N or C-terminus then you don't need capping and can keep it in the charged state. The capping choice is a matter of the force-field/simulation software you're using, not the protein. AMBER ffs have the ACE/NME which you can manually build onto the termini in a software like ChimeraX. The residue and atom naming if using ACE/NME must match the ff files (.rtp files for e.g.). I believe if using C36m then one can use gmx termini caps. Read some methods sections of simulations that are similar to your system and what you want to do to identify best practices.

2. This is entirely dependent upon your expected outcomes and what you want to show with your simulations. I would think carefully about the impact of removing such a large disordered region on the conformational dynamics of the receptor core. Capping would seem appropriate, otherwise you would have two dangling charges. However, either way there are going to be non-native interactions between the charged ends and the helical core.

I would read papers that are strong in their methods and similar to your system to understand the best approach. I hope this helps.