He used Maul as bait to lure out his mother and kill her, and when Maul escaped Sidious no longer considered him a real threat. Maul's hatred and bitterness against Sidious (and the Sith as a whole) spurred him to find ancient Sith artifacts to use to exact his revenge.
During the days of Star Wars: Revenge of the Sith, Maul had a deep hatred towards Palpatine and wanted to tear down his plans. So, it would seem logical to publicly expose his old master. But two things were standing in Maul's way: he was beyond terrified of Palpatine and knew the Jedi wouldn't trust him.
Ultimately though, Maul remained driven by the desire for revenge, specifically against Sidious, the Master who disowned him, and Kenobi, the Jedi who defeated him.
As Darth Sidious, Palpatine took Maul as an infant from his mother Talzin. While still apprenticed to his own master Darth Plagieus, Sidious trained the Nightbrother in the ways of the Sith. Maul was originally loyal to Sidious, obeying his every command without question.
Sidious realizes the infant is strong in the Force, and would become a threat if found by the Jedi. Concealing the existence of his own master, Sidious raises Maul to believe that he is a Sith apprentice, but he actually intends him to be an expendable minion rather than an heir.
He regretted it because he had essentially wasted 20 years training him. But since he was an enemy at the time of his death on Tatooine, Sid was glad that he did not have to keep watching his back anymore now that the loose ends were tied off.
Regardless, based on the season 7 of The Clone Wars, and the fact that Darth Maul was aware of Anakin Skywalker's future and his role as the Chosen One, it is very likely that Darth Maul indeed feared Darth Vader to some extent, and preferred to avoid a direct confrontation with him, at least, on his own.
Dooku despises Maul for being a threat to him, and for killing his beloved padawan Qui-Gon, and Maul hated Dooku for being his replacement. They also didn't regard each other as a true Sith: Dooku saw Maul as nothing more than a trained animal and Maul saw Dooku as a Sith pretender.
He was after Obi-Wan for a long time but realizes that Palpatine was the cause of his problems. Maul was blinded by his hate for Kenobi to realize Palpatine was his biggest threat all along. The true answer is himself.
And while his warnings don't work, at the very least, it shows that Maul became aware of Anakin and what he would soon become. With the rise of the Empire, it's likely Maul caught on to Vader's existence pretty quickly.
Thought dead, Darth Maul survived his injuries by focusing on his hatred of Obi-Wan Kenobi, the Jedi who cut him in half. His shattered body was dumped amid the refuse of the junk planet Lotho Minor, where the once deadly warrior fell into madness, staying alive on a diet of vermin.
In the novelization he seems to feel… mild regret, maybe? He didn't expect Maul to die, and this might indicate a degree of respect for his apprentice and his skills; he didn't hold it against Maul that he lost, but honestly believed that the Jedi would be killed, so he might have felt that he underestimated them.
Maul most likely knew Sidious' double role as Palpatine and Sidious when he served as Palpatine's apprentice. He definitely knew about Palpatine's goal of creating a galactic empire ruled by the Sith. Maul knows that Sidious is the Emperor.
Maul also never openly opposed Sidious. He became a rival in a way, but he knew he never really stood a chance of overthrowing him, specially after his injury. Palpatine had so much power and control over the senate.
Like his master, Darth Sidious, better known as Sheev Palpatine, Maul's main weakness was his overconfidence. In Luke Skywalker's confrontation with Palpatine during The Return of the Jedi, he tells the Emperor that his overconfidence is his weakness.
Following Qui-Gon's death by Maul's hand on Naboo, Dooku visited Palpatine in distress about his apprentice's death, hoping to have one day made Jinn an ally while also directly stating he knew of Maul's existence.
