Are patented seeds of sweet seedless peppers from breedx.com GMO?

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.

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;

1. I really don't understand why you think it wouldn't be. There are some sources recommending that boliological waste made up of the GMOs themselves be sterilized before leaving lab conditions but if you eat a GMO and it passes through your digestive track there will be few if any living GMO cells remaining. Particularly in the case of peppers, mammals' digestive tracts will destroy pepper seeds. That's why they're spicy, it's ironically a defense mechanism to keep us mammals from eating them.

2. At any rate, 1is kind of a moot point because the paper you linked clearly states that wild peppers were cross bred with commercial peppers. That's very traditional plant breeding, no mention of GMOs. Given the blatant fear mongering in the rest of the paper, I'd be floored if they missed a chance to scare people about GMOs in these peppers. So unless the peppers you're asking about are different from the ones in the paper, I'd say they're definitely not GMOs. Also, I don't believe there are any GMO peppers on the market at present.

3. The short version is this. A company, let's say Pioneer seed, patents a breed of corn that has, let's say increased stalk strength for windstorm prone areas. A farmer buys and plants those seeds, sells the resulting crop. The only difference from heirloom seeds is that the farmer is legally prohibited from using that crop as seed corn and selling that crop.

4. So in principle, there isn't really an impact on society from patented seeds. In practice, some of the patent holders have been overly aggressive with there enforcement. IMHO, this is a patent enforcement issue not an issue with the parents themselves. I don't know about Europe but I know that here in the US there is a problem with dubious patents being approved and enforced but again, that's patents as a whole not just seed patents. At this point I'd be more worried about what happens without seed patents. Nobody is going develop seeds except universitys which (at least here in the US) are criminally underfunded. Effectively, our crop technology would stagnant without serious increases in public University funding which I'm a huge supporter of but sadly, can't imagine happening in my lifetime.

I hope I'm not coming off as an asshole here. Just trying to answer your questions honestly.