Does Syd have Rogue's powers?
No. As we saw last week, her power is that when someone touches her, they switch bodies. At first I thought they switched minds, but considering the way they switched back, I think their bodies physically transpose (and bring their powers with them), and the reason nobody realized that the first time is because it happened during a confusing melee where it was hard to keep track of who was where.
I hadn't expected David and Lenny seemingly to know each other prior to the Clockworks Psychiatric Hospital. Makes me wonder about the circumstances of their admittance into the hospital and whether they were admitted individually or not.
Or were those memories even real? Remember, Ptonomy said that if you interact with someone in a relived memory, it changes the memory. Maybe the ghost of Lenny living in his brain is putting herself in his memories and rewriting them in the process. (Or himself? Apparently the character was written as a dirty old man and Aubrey Plaza refused to let them rewrite the lines for a woman. So it's unclear how the character prefers to be gendered. Maybe I should say "themself." Themselves?)
I also wonder if the Devil with Yellow Eyes exists entirely in David's head (but as a living creature) and somehow arrived there when David "heard" the stars.
I found it interesting that he couldn't even bring himself to say the word "star" or "astronomer" to the therapist. There's some kind of traumatic block there, I think.
I suspect he's a stepfather and not actually Charles.
This show is not set in the X-Men universe per se; it's taking the David Haller character and doing something distinct with him. So this version of David may not be the son of Charles Xavier at all -- just as the Marvel Cinematic Universe version of Star-Lord has been given a different father from the comics' version, and just as the movie version of Havok is Cyclops's older brother instead of younger, and so on.
There's definitely some mystery surrounding David's father in the show, so he will be an important character, but I think the producers want to avoid using Charles Xavier. So they'll probably come up with a different but similarly powerful character to be David's father.
By the way, it was weird that the receptionist at the hospital addressed David's sister as "Mrs. Haller." Haller is her birth name, her brother's surname, so it wouldn't be her married name.