I am currently with another host. As part of their service, they offer web design and templates for pages etc. If I switch hosts, will I be able design there and host here or is there similar service with DH?
If you can still access your account there and FTP download the files to home, then you can design there and host here.
What DH offers here are One-Click installs of software like WordPress, where you can install themes. DH doesn’t offer templates or web design for static sites. Personally, I use WordPress for a handful of sites (some blog, and some with regular content) and tailor the site to my tastes.
I think I know what you are getting from your current hosting company. What they offered you is something like a website builder. I don’t think you can download the source files because they are hosted in their server. You only have access to edit some of the contents on the templates.
If the above is correct, you will lose all your html files if you switch to DH host. If you are able to download the files, you can simply re-upload them to your DH account.
If it is a static html you can always view the html, then copy/paste and upload it to DH.
Careful with that, though. If your previous host was using a web site template system, there’s a significant chance that the templates are probably not yours to keep when you leave.