I’m a 17.4 HCP and my driver swing speed is 105-110. I’ve gotten as high as 113 after warmup and really going after it. I’m getting numbers from a GCQ launch monitor so I believe they're reliable, and on course distances back this up. Yesterday I had a drive carry 267 to the green of a short par 4. I think this carry distance is more or less aligned with a swing speed of 105-110 but I’m not an expert in this data. My ball striking is on par with my HCP so I don’t often get the ideal theoretical carry for my swing speed. I played baseball as a kid, and have been taking swing lessons for over a year which has increased my speed a bit from when I started.
Unless money is an issue, I would recommend changing driver. It is easier to get a club fit for your swing than it is to change your swing. Please remember that the amount of time you need to get a new swing perfected is long indeed. Even the best pro's struggle for a while when they change their swing, and very few amateurs have the time, patience or talent to change their swing. That being said, do consider changing your swing if your current one is very bad.
Not entirely. The length of your shaft affects club head speed, and the head type itself can help with the launch angle. There is a reason that as your swing speed slows down, your longer irons don't go anywhere, and that is the launch angle. You will probably find with a bit of practice that you can get the 4H to have better distance and consistency than your 5i.
Not impossible, but the odds are fairly low. There are multiple guys on the LDA who have very unorthodox swings, can't really play golf, but can hit the ball a long way. Realistically, most of these guys are exaggerating, though the odd one will be capable of it.