To my knowledge imgur is about the best there is in this class right now. Usage is free and while you can link to a page that includes the image along with comments and imgur site features, you can also direct link to the image and even embed in other sites.
Image links are reasonably short by default. I disagree with your version of what makes a url "scary" (in that real scary links often have intentionally harmless names that mask their nefarious purpose) and using a hashed id value should not in itself be a scary thing. At the same time, if you have an imgur account you can set custom urls for your resources as well as locate them on your own sub-domain.
And yes, if you upload from your account you may later remove images from circulation.
I like Dropbox for that. I most use it for screenshot sharing really really really easily and fast (print screen, dropbox (desktop) app automatically saves the screenshot, right-click the app and then copy the sharing url to the new file).
Free: Yes (well up to current free storage limit - currently 2gb but exandable for zero monetary cost through some actions).
Images are accessible via a direct link: Yes
Link is reasonably short: hmm well that depends on what reasonably is defined as; an example would be: dropbox.com/s/kzil0355ienrzko/Screenshot 2014-05-14 09.41.21.png - (the last part is just the filename you choose).
Viewers see no advertisement. Uploader can be shown advertisement: No adverts for anyone
Link is not too scary (imageshare.com/pic-of-my-cat.jpg is OK, but bit.ly/3yt9e53 is scary): well that's somewhat subjective... see above for an example. Yes it has a hash type segment but the last part is user settable (as it is the filename).
Logging in before uploading allows you to remove a picture in the future: Yes removal at any time is possible. (Note: with Dropbox you must login to upload.