As far as i know a website can be accessed on
and conditions
Is it possible to make our site's content at least accessible in all conditions?
Should we think about all conditions?
Is it possible to make our site's content at least accessible in all conditions?
Yes. But the more complicated and feature-rich the site becomes, the harder it gets to make it accessible.
A plain-text site will be accessible to nearly everybody, a website built in Flash only to a smaller group (and less accessible even to that group, but that's a different story).
Should we think about all conditions?
This depends entirely on your target audience, and the amount of time and money you're willing to invest.
For a government or a health agency, accessibility is very important. For a website with Flash games, probably less so.
Look at the stats of your website, see what combinations of screens your users use, and spend time only on the common ones (>2%).
People with B/W screens (is there such a thing?), text-only browsers, and (CSS or JS) disabled deserve a less then optimal experience, so don't waste time on them.
Is it possible to make our site's content at least accessible in all conditions?
Yes it is. What makes HTML well designed for this is that the proper use of semantic mark-up means that you don't have to think about individually all those cases that you listed.
I find it useful to consider the most difficult cases. If the user has failing sight and hearing and needs or wants to use a Braille device to browse the web, can they use your web site? If the user is quadriplegic and needs to navigate web pages using a sip-and-puff input device, can they use your web site? In each case, does your web site provide them with the best experience that they are capable of receiving?
Cover those cases, add a few basics like ensuring adequate colour contrast, make your web site as functionally rich and attractive as you wish for your average users, and you should find that support for the cases you list can be left to the devices themselves to provide appropriate accessibility support.
Read, understand and use WCAG 2.0 and WAI-ARIA. You will gain a much better understanding of the techniques you need than by asking individual questions on SO.