If you're referring to the editor in the ACP then the possibilities are endless (almost) as you can edit the Style's HTML and CSS from there. Having said that it doesn't actually rewrite the file, just creates a cache so if you happen to clear your style cache for one reason or another you'll lose all those changes - that's why it's recommenended you actually download / edit / upload the individual HTML / CSS files.
Of course, if you're on a free host which you are...then you'd have to ask on your support forum. ~neutral~
Your first point is not actually true... It does write the files on the server aslong as the style folders permissions are right and that in details on the template page the templates are stored in the file system NOT the db. If they are stored in the db it stores all the changes to templates in the db so when you hit refresh it grabs all the files from the server which don't contain these changes. The same goes with the css. You need to ensure that it is reading and storing the changes from the file system.