Hi OP,
We did this around two years ago. There's a very long thread on here somewhere from me about the process of building the annexe/extending the main house.
Re deprivation of assets - this wouldn't definitely be the case, if your parents needed care. You need a very good financial advisor who can set out clearly the reasons that you're entering into this arrangement. It's important to do this before there are any signs that your parents need care that you can't provide. If you are able to show that there were sound reasons for the decision to move in together, and that they were in good health at the time with no reason to suspect they would need care, deprivation of assets shouldn't apply.
Re living in houses nearby rather than an annexe - this really depends on your relationship. My DM has cerebral palsy and I know for her, it's a massive relief to know that I'm literally in the next room. Also, if I'm having a lazy day, I can pop into her home in my PJs and barefeet. I've got two disabled DC with high needs - the logistics of having to go down the street would make it a faff. Next door means it's much easier just to pop in whenever you need.
Re privacy - you need open, honest, and frank discussions. You need very clear boundaries to prevent resentment on either side. Are there any occasions you don't share with your parents at the moment - Christmas? Birthdays? Easter? You need to think about how it would feel to exclude them when they are literally next door. And then you need to think about how it would feel to have no family time on big occasions that doesn't include them too. You need a balance but this requires the ability to have open and honest conversations about what you all need/want. Not easy!
Re future caring - you need to be clear about what future care you are willing/able to provide and at what point you'd need to consider a care home/carers. My DM is adamant she won't ever consider a care home. I also cared for my dad until he died so I'm OK with providing that role. It can be demanding though. My DM has just been diagnosed with early dementia, in addition to her cerebral palsy, so it's already hard work at times.
Re practicalities of living - we bought a wide house, which is also a corner plot. This meant we were able to section the house off to the side and DM has her own garden and we have ours. We also moved the kitchen and living rooms around so we weren't either side of the same door. The internal connecting door now runs from our hallway to the study then laundry room, and then the internal door. This provides us both with privacy to chat without the other side overhearing. I didn't realise how important this would be!
Re the practicalities of building - this is what my previous thread was about. We opted for a modular build because it's 1) super fast 2) cheaper than standard construction 3) extremely energy efficient 4) feels just like a "normal" house! It was up within a couple of months. We had originally planned a traditional construction but I'm glad we went via the modular route.
I am glad we made this move as it makes it easier for me to provide support to DM (and my stepdad) without eating up huge chunks of my time travelling back and forth.
If there's anything specific you want to know, please ask. I've tried to think of what I'd find useful if I was doing this all over again!