Is it possible to do a DNS redirect so that mywebsite.com would remap to MyWebsite.com?
This would be for purely cosmetic purposes on the domain name alone. I understand that the domain name will ultimately resolve to a lower case version, and that all characters following the TLD are best kept in lowercase.
What I’d like to achieve is simply maintaining MyWebsite.com/whatever-in-lower-case in the URL bar.
DNS is not case-sensitive, so this is not likely to be possible. You can modify your bookmarks/home page links/embedded URLs/whatever to contain the case you want, though, but I’m not sure it’s really worth the effort…