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It’s useful to use Bitbucket since they offer free private repositories. So instead of just having a local repo, you can also have another remote one. It’s good for redundancy purposes.
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Yes your host would need to have Git to do a straight deploy with Git. But you could also deploy via FTP/SFTP but using a Git repository. This means that you’ll actually transfer files via FTP/SFTP (which I’m sure your host supports) but only transfer the files in your Git repo. That means not including files that are git ignored. This is still a win since you aren’t doing it manually. Here’s a Grunt plugin for it: https://github.com/thrashr888/grunt-sftp-deploy
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No, an IDE can be useful for interacting with a Git repository but they probably can’t do those commands easily.