We have seen the singularity brought Jay Garrick to this Earth without his powers
No we haven't. Nothing about the singularity was stated to take powers away. We later hear Jay say Zoom took his powers, which he later admitted was a lie, and that he lost his powers due to using a speed enhancing formula similar to Velocity-6.
How is Dr. Zoom able to come to Barry's earth with power?
See above. Also, just "Zoom", not called Dr. Zoom in the show.
If any channel exists between the two earths, then why is only one way travel shown?
The main singularity allowed two way travel. Zoom is obviously returning to Earth 2 as well. The scoobies had to stabilize the singularity in Star Labs to allow travel to Earth 2 using the Speed Cannon, so the same needs to be done on that end, for normal people.
They sent Grood and the Trickster's Bombs via the Speed Cannon. Zoom can jump through breeches without issue and without a Speed Cannon, likely due to his faster speed, unlike Barry.
How does Dr. Zoom know about this speedster on earth-one (Barry's world)?
Not explained yet. Tune in next week(s). Likely he did the same as Jay and Wells-2, and did surveillance/research. Or picked up a newspaper. Most likely, he used Earth-2 Cisco, the villainous Reverb, to find out.
How did Harry (earth-two) know about Mercury Labs and why does he steal an unknown device?
Not explained yet. Tune in next week(s). Likely, Mercury Labs exists on Earth-2.
If Doctor Zoom is so powerful then why is he sending meta-humans to kill Flash rather than coming himself?
Explained in the Christmas episode:
Zoom wants to fatten Barry up like a Speed-Force Turkey before taking his power.