The unwritable directory issue is a bug I am aware of for a week and I am publishing a new release today to fix it. I delayed the release because the bug is not critical; you can still skip through the first page, restore the database and in the third page (Server Setup) you can simply check the "Use directories under the new site's root" box to have ABI itself automatically fix the problem. Downgrading to 3.3.3 is not a very good solution. 3.3.3 has some bugs which could come back to bite you. 3.3.4 would be a much better choice or you can just wait one more hour for 3.3.7 to be released.

Regarding the AJAX error, you missed the obvious: our troubleshooter! A link to the troubleshooter page is given to you by the Akeeba Backup Installer itself when it detects an error. ABI also provides you a link to the documentation page -with the same information!- on the top of the page, inside the big green box. One of the points made is that you need to use only alphanumeric (a-z, A-Z, 0-9) characters.

This is not a limitation or bug in Akeeba Backup Installer. The problem comes from your host's Apache setup, namely the mod_security2 ruleset. The servers are set up to filter potentially dangerous and suspect query parameters. A password with special characters like a hash sign, dollar sign, percent sign, caret and star will most likely trigger that protection, throwing a 406 (Not Accepted) error. Since most servers do not have a server error document for the extremely rare 406 error, you also get a 404 error (Not Found) when Apache tries to retrieve the error document for the real error, the 406 error. Of course, all of that makes it completely impossible for a regular human to figure out what the heck is going on. It took me a lot of digging and the help of my host's technicians to confirm that this is a mod_security2 issue.

Anyway, you posted the request while all of us were off duty and couldn't help you. We apologise for the inconvenience and please accept a free upgrade to the AKEEBAPRO subscription level

Thank you for your kind words, Nancy! If the school site also needs anti-hacker protection, I am willing to donate an ATPRO (Admin Tools Pro) subscription as well. I have a soft spot when it comes to non-profits helping people get educated.

Well, since you have a backup with 3.3.3, you can use it to restore your site. The bugs which existed in 3.3.3 prevented some sites from backing up at all. Once the backup is restored, please install Akeeba Backup Professional 3.3.7's ZIP package (released just 12 hours ago) on your site twice, without uninstalling 3.3.3 before doing that. The "twice" part is required due to a known issue in the Joomla! extension installer which causes some files to go missing the first time you install the ZIP package. 3.3.7 fixes the bug you ran into and should work perfectly.