Are patented seeds of sweet seedless peppers from breedx.com GMO?
You as a consumer will not ever buy GMO seeds, accidentally or intentionally. Because the genome is a protected product, farms who buy GMO seeds from companies like Bayer (formerly Monsanto) have to enter into a legal agreement with the seed supplier, and they buy massive quantities at a time. Many public seed companies proudly declare their seeds are non-GMO, but that's true of all seeds you'd be purchasing.
The seedless plants that you as a consumer can buy are bred by creating a sterile hybrid between two non-sterile lineages. It's essentially a "defect" in the children of the two lineages which prevents their progeny from developing seeds even though they still develop fruit.
Edit to answer the rest of your questions:
Legal to use biological waste: Use freely.
How the patents work: Patented plants are basically just a legal protection for the company that produces the seeds you're buying. They've put a lot of work into generating lineages of pepper plants which can be cross-bred to produce seedless peppers, and their patent ensures that they are the only legal supplier of these plants (these specific plants--someone else could breed separate lineages and patent their plants without any issue). The USDA website and US Patent and Trademark Office website have more information, but I'll summarize: You could be sued if you bought their patented seeds, grew pepper plants from those seeds, then created a business to propagate and sell those pepper plants. You, at home, growing food for you and your family/friends? No one cares. The patent only exists to prevent another company from taking the plant that the original company painstakingly bred and selling it as their own.
Implications for society: You can't build a business selling their patented plants without a licensing agreement, I guess. Nothing odious about hybrids, and protecting specially-bred plants is enshrined in the Plant Patent Act of 1930, so it's been around a long long time.
According to their website, no.
All our varieties are bred using classic and traditional seed breeding techniques without any genetic modification (non-GMO).
"is it legal to use biological waste after consuming those peppers?"
I do not understand this question.
"is is healthy?"
To quote Bill Clinton: "it depends on what the meaning of ‘is’ is"
So, while in think there are certainly fair criticisms to be made of allowing patents on plants, the paper you linked is kind of just low quality fear mongering. It's heavy one scare tactics and light on facts. I wouldn't let anything in this paper keep you up at night without verifying it through a more reputable source.
To try to answer your questions though;
I hope I'm not coming off as an asshole here. Just trying to answer your questions honestly.
But if I take unconsumed food from my plate and use it for compost or just dispose it in garden? Is that a patent use legal issue? Should laws protect us from unwanted GMO patents? For example, if I have a field and field next to mine is growing GMO, why I would be in danger of legal issues? I should get protection for using my land for growing what I need.
If it is only crossing and breeding, how that can be patented? How can you prove that I didn't use protected patents?
GMO isn't bad. Everything is genetically modified.
Patented foods already exist. Have for years.
Monsanto has corn plants that don't grow at a consistent height if you try to replant the seeds, making them profit by getting you to buy more seeds next year.
We already have seedless plants, bananas.
Just wait until they hear about this one fruit called an "apple".
Like most things, the real problem behind GMO is greed. Creating rice strains that grow in impoverished areas, where little else will grow, is hard to see as a bad thing. We could be, and to some degree are, creating strains to solve world hunger, improve nutrition, improve durability of produce without sacrificing flavor. Tomatoes, I'm looking at you.
But so much of GMO is an effort to dominate the market, instead of to make the market better.