There is no web viewable PhotoStream except for ones you choose to share with others. You can access your pictures from iOS devices (iPhone, iPad, iPod touch and AppleTV) plus Macs and PCs.
Photos aren't the only thing that uses iCloud space, so there are many reasons why you'd purchase extra.
After signing into your iCloud account on your iPad, your data stored in iCloud (such as contacts, calendars, notes, email, bookmarks etc.) automatically appears in the various apps that iCloud uses, such as Contacts, Calendars, Safari, Notes, Photos.
It sounds like familiarising yourself with what iCloud does, how it works and how to use it would be beneficial to you. I'd recommend reading the Help:
I tried "sharing" my photos with myself through my home email address - and when I went to the link, it was a bunch of spaces where images belong with an exclamation point. My home computers are macs. Just older ones. I have both a laptop and a tower.
My ipad (also older) was able to see the web link and the photo stream came through.
Still not clear why they would make this unaccessible to their own customers.
If the images were blank, maybe you hadn't waited long enough for the stream to be uploaded/created.
PhotoStream is designed as a way of wirelessly transferring the last 1000 photos taken on your iOS devices to your other iOS devices, for up to 30 days, plus the ability to download them on your computer for permanent storage. It requires OS X 10.7.4 at least to be accessible on a Mac.
It is not designed to be an online web viewable photo gallery. If that is the type of service you want, then there are literally dozens of such services available.