First sale doctrine. The first sale doctrine provides the right "to sell or otherwise dispose" of authorized copies of copyrighted works so as a general rule you will probably be okay using fabric that includes copyrighted images (assuming it's not bootleg fabric). There are limitations and complications. The more you split up a purchased work -- for example cutting out prints from a photography book and reselling them as framed works -- the more likely someone will come after you. We explained these principles here and here.
Copying from books. You will not be okay copying images from books or photos and applying them to coffee sleeves. In that case you're not reusing a copyrighted work, you're actually making copies and that's prohibitedunder copyright law.
"Intended for Personal Use." We wouldn't worry too much about statements such as "Intended for Personal Use" unless as a condition of buying fabric or some other product, you entered into a license. That fact should have been evident at the time of purchase -- often a statement such as "By breaking the seal on this package, you are entering into a license, etc. "
Using trademarks. Even if you purchased authorized fabric, you're almost always violating the law by placing another company's trademarks on your product because it is likely to confuse consumers that the company that owns the trademark is associated with your venture. Consider the situation from the trademark owner's perspective. Would you want your mark to appear on a product that you didn't make, had no control over, and for which you receive no revenue?
Making a good faith effort. Making a good faith effort to find the owner of rights doesn't get you off the hook for infringement but it is likely to reduce the damages if someone comes after you. For this reason you should always document your searches.
Blah blah blah Dept. As for your question as to why (or whether) the law protects corporate entities with huge bankrolls ... the Dear Rich staff believes that -- considering the terrible governments and laws that exist in the world today -- the U.S. copyright law and our federal judges are relatively fair and balanced. We think your beef is with the way the legal system (not "the law") can be gamed by fat cats because the well-heeled can better survive the costs of legal enforcement. In truth, digital copying of movies, music and books has reduced many of these "huge bankrolls" (consider the demise of EMI for example).