Guest Relations inside the parks have long counters. As the other poster mentioned, the Studio has some small 'pod' desks. Guests are asked to wait in line in an area away from the desk until they get to the front and there is a CM free. That gives more privacy when you are talking to the CM. The stations are several feet apart and the CMs are usually spread out a bit.
If it is busy, there is also a CM talking to people in line - some people are just looking for things that can be taken care of easily, like maps, buttons or a piece of information.
Sometimes, a guest requesting
DAS will be pulled to the side, outside of the line to deal with.
You can bring a stroller in to Guest Relations at all parks. MK has stairs to the front door, but there are ramps to doors on each side. The others are at ground level.
Your family don't have to be right there while you are talking to the CM. They will need to have their Magicbands or tickets scanned as part of the process of linking those to your the DAS to your ticket or Magicband. They don't necessarily be up by you if the DAS is for you.
You don't need to talk about your medical condition or go into great detail. The CMs need to know what your issues or concerns with waiting in the actual line are, not your medical condition or things that are not related to waiting in line.
If you haven't seen it, there is a WDW DAS thread stuck near the top of this board. The actual thread is from before and right after DAS was put into place. The first post is kept up to date, but don't bother reading the other posts.