They don't. They just convert it to JPG.
There was a partial outage with Imgur's delivery network (Edgecast). Everything is back up and running though.
Thanks for reporting this. It's been fixed.
That option is only available from the account section. It's not available from the home screen because of the massive load it takes to resize lots of images at once. So, you need an account.
Yes it is, click on the generate links link.
Hyperdesktop is not made by Imgur and has no association with Imgur other than it being linked to in the Tools section. We can't control whether or not it works. Besides, I have it installed on several computers and have not had any trouble with it. Do you have the latest version?
You'll have to upload the image and then make a second call to place it into an album.
Sure it is. Just check "Edit" once you've selected your images to upload. You will be taken to the edit image page and there's a resizing option in the top right corner.
Nope, it does not. API keys are only used on upload.
Sure, you can do that anonymously. No need to be signed in unless you want to use /2/account calls.