Permissions can work like that, the reason 777 shows as a warning is because those permissions are bit more open than necessary. 755 is the recommended as that gives high enough access that BackupBuddy needs and not being as "loose" that 777 is. Permissions can be set in FTP though some hosts restrict some places you can change permissions yourself so for this can ask host about.
Windows servers can work with BackupBuddy, Linux servers are preferred but not required. Something on Windows servers that can help when using them is getting the zip.exe
set up. Without it then BackupBuddy tries to run in compatibility mode which happens to be the other warning you got. BackupBuddy can work using the slower zip methods but not preferred as the bigger the site is then the more likely it runs into an issue with it. Getting the zip.exe set up on the server should let BackupBuddy run in the faster mode so that is good. Most likely will need to ask host to help set up the zip.exe as just putting the file in the web root isn't quite enough for most places, the server needs to be told to look for the zip.exe and to use as it won't be default if just uploading the file to the site.
So yes, BackupBuddy can work on your site as long as the site is not really really large. Getting command line zip will really help though.
Welcome to the forums. Be sure to post your questions in the correct forum and don't forget about the new and improved search function located in the top right of the forum pages. The gear/wheel symbol next to the search box will give you more options for advanced searching. Thank you and have a nice day.