Welcome to the forum.
1. That may depend on your company. There are certain insurances, etc, that employers pay here so you may need to talk to them about that. Alternatively, you might be able to be self-employed or be paid through a Swiss agency that handles all that sort of stuff. Do a search on the forum for "working remotely", "self-employment" and "ANOBAG" to see what's been said recently.
2. Yes, you need to register because you will be a resident here. It doesn't matter whether you're working or not; you're going to be living here so you must register.
3. The canton will issue you with a permit; what type will depend on how you come here. For example, your boyfriend might be able to apply for a concubine permit for you which means you'd get the same type of permit he does. Not all cantons offer this permit though so he'd need to check with the cantonal migration office and if they do issue the permit he'd have to agree to be financially responsible for you for 5 years.
4. Again, this will depend on how you come here. See above. Otherwise, it will depend on what type of permit you do get.
5. Yes, you need Swiss health insurance cover within 3 months of arriving.