Hi Poli,
The main photo storage/sharing sites are listed at
http://www.techlicious.com/guide/best-photo-sharing-sites/. Not all sites offer photo-editing features. There also other general file storage websites like Dropbox, Box, etc. where you can can create folders and upload and download photo's (as well as other files). But websites in this category are not specifically geared to photo and don't have photo editors as far as I know.
By the sound of it GooglePhotos would seem a good option: almost unlimited storage space and a free, easy-to-use editor online to crop and adjust your photo's. Flickr might be a good alternative (storage + editing). At the moment the Flickr photo editor in unavailable. No idea for how long.
Wherever you store your photo's you'll need to create an account and sign in at least once. Most sites have a 'remember me' box on their sign-in sites so that you're then signed in automatically whenever you visit the website. So you can create a link to the website on your desktop, for example. Clicking on this link takes you to the website where you're automatically signed-in and can view, edit and upload photos.
But write the sign-in password down somewhere. If at some point, someone cleans up the 'cookies' on your desktop, the automatic sign-in won't work until you sign-in again and click on the 'remember me' box.
With GooglePhotos and Flickr you can set the 'sharing options' for all or some albums and even for individual photo's. The most usual option is 'people who I send a link to can view but not edit my photos'.
Hope this helps,
Mike