Professor Paradox was a human involved in experimentation involving quartz crystals in the 1950's. The name of the project was Paradox. He has found special qualities in the crystals. It was a government funded project mainly aimed at creating a wormhole. It was largely a success. But a slight miscalculation by the professor and an accident caused by his assistant Hugo turned the professor into a timeless being and threw him across event horizon, while turning Hugo into an extra-dimensional creature wreaking havoc across time continuum.
For about a hundred thousand years, Professor Paradox remained in a reality outside of time, where he didn't need to eat or sleep. He just existed. At first, he went insane. But then, as he said, he simply got bored of insanity and went sane again. It is here that Professor Paradox gains unrivaled knowledge on the workings of space-time continuum, and a complete understanding of it. This gave the professor some of his wildly exhibited abilities such as his teleportation abilities.
Learning that the accident that gave rise to him also gave birth to an extra-dimensional creature that arrives 50 years into the timeline, Paradox goes to the particular timeline (Alien Force) where Ben 10 is a teen. He takes Ben and company to the particular timeline of the accident on his request, where he figures out that Hugo was the creature. We also see an alternate Paradox from a different timeline, meaning more than a single Paradox exists, but not outside of the knowledge of each other. They work as a singularity even though they are not. Professor Paradox then takes his former apprentice on time-traveling on his request.
Paradox later appears in the Highbreed invasion arc finale to assist Azmuth from being annihilated. He also assists Ben 10 in his fight against the Highbreeds, although on a relatively minor scale. Should he so chose, Professor Paradox is more than capable of making sure the Highbreed never existed. But his role is more along the lines of making sure balance exists in the multiverse.
Paradox also appears in the episode where Gwen (an Anodite), meddles with time and reality. Paradox first appears to warn Gwen. Later he appears to tell Gwen what her meddling has caused.
Paradox appears in Ultimate Alien as the creator of the Map of Infinity and planet Perplexahedron. Paradox appears in the Forge of Creation to help Ben 10 stop Ultimate Aggregor from absorbing the power of a baby Celestialsapien. He helps a young Ben 10 return to his own timeline.
It is revealed that Paradox is forbidden from coming anywhere near 500 lightyears of a Celestialsapien or Forge of Creation. However, the reason why this is so has never been revealed.
Paradox later appears Ben 10: Omniverse. In an unknown time in his life, Professor Paradox has been involved in a Time War (war involving Chronosapiens gone rogue. Most beings in the universe are not even aware of such a war). During the timeline, Paradox had to disguise his Chrononavigator (his pocket watch), for safe keeping. The device could alter space, time and reality itself. This was displayed when Eon got his hands on the device.
Teleportation/Time Travel: Professor Paradox can teleport through time, and space effortlessly, across planets, galaxies and universes respectively, in a matter of seconds. He has a map of the multiverse in his mind that he uses. While he can effortlessly "walk" to appear anywhere, he needs to use his Chrononavigator for time-travel.
Time Manipulation: Professor Paradox can slow down or accelerate time, that of his own, an object or the entirety of Earth or anything if he so chooses. He displays this ability in the fight against Highbreeds where he took down a bunch of DNAliens while appearing to be unbelievably fast, but on his own, the professor was only walking. On an another instance, Professor Paradox takes an old Kevin (from having come into contact with a time accelerating extra dimensional creature), and returns with him, turned back into a teen.
Timeless/Immortality: Professor Paradox ceased to age when he gets sent to the event horizon. He does not need to eat or sleep. He is immortal in all sense of the word. This was seen when Paradox used himself to momentarily stop Hugo (the extra-dimensional creature version of him). Merely coming into contact with the creature aged Kevin to his 60's while Paradox seemed unharmed. He commented that he could feel eons passing through him but there was no obvious change in his appearance.
Forced Transformation: Although we have no clear indication of the exact ability, Professor Paradox transformed Ben 10 from his Alien X form back to his human form. This implies, he has the power to manipulate existence. It wouldn't be the first time either. Professor Paradox with ease, made it possible for a human form Ben and Gwen to breathe on Moon, and in the space inside the Forge of Creation (the extra dimension & home of Celestial Sapiens). This also implies Professor Paradox has control over space & time to a certain degree.
Professor Paradox's most obvious weakness is Celestialsapiens. Since Paradox has total understanding of space-time continuum, the ability to go up or down the aspects of time, it could be argued that his power rivals that of the celestialsapiens. For this very reason, Paradox is forbidden from being anywhere near 500 lightyears of one, up or down the time continuum. Since Paradox is tasked, by himself or not; to keep a balance in time, Paradox simply has to revisit, retry, undo or whatever necessary to fix that which has gone wrong. The paradoxes in repeat sequence is perhaps Professor Paradox's biggest and most obvious weakness, since he could essentially be stuck doing the same thing for eternity. Time does not apply to him, so he is stuck doing what he has been doing over and over. He often ends up creating alternate versions of himself doing the same sequencing over and over.