It depends. Most hosting providers will either discourage you or prevent you from relaying messages from your servers, so that is something you need to check for. Also, you'll want to make sure that your dedicated IP is persistent and won't be lost across reboots. Once you establish a good sending reputation, that's valuable in making sure your messages reach the Inbox.

In the case of Mailgun, you should be assigned an IP with a neutral reputation. For example, before dedicated IPs are reassigned we leave them dormant for at least a month, usually much longer, before assigning to a new customer.

Paraphrasing Oscar Wilde, every IP address has a past; in theory, an email service will ensure that the IP they give you has a clean bill of health (not marked in blacklists and such), whereas your ISP or VPS provider may not.

That depends on who you are.

Mailgun will do some chores for you, for a fee. If those chores are things you can do in your sleep, fine! Mailgun does nothing magic.