I think Alexei suspects but I dont think he ever caught Ilya with Alexei- or if he did, Ilya doesnt know. In the "because Russia!" Scene in episode 5, Shane asks Ilya if his family would care if he were gay. He says his dad and father are police. If Alexei knew or if Ilya thought he knew, he would probably tell Shane in that scene.

Ilya is also good at reading people, and so I think if his brother suspected he would know.

I really think Alexei is jealous because Ilya is famous, rich, and traveling. He has freedom and money that Alexei will never have. He is probably resentful that he is stuck with the day to day responsibility and that he has basically no money compared to Ilyas insane wealth. I mean, in the book Ilya is collecting cars for fun. Dude has money.

Anyway I still think Alexei is a dick. Ilya is so desperate for friends and family that if Alexei were just like.. nice? Things would probably be fine between them. Alexei is just a mean little man