Hi guys
I am pretty new to the frequent flying community.

Regarding the Global Entry/TSA-Pre I have the following questions that I am unable to find the answer to on regular websites:

1. I am on H1B visa here in US and I want to register for TSA-Pre. Am I eligible to do so without going through Global Entry?

2. I am planning to go back to India for a couple of weeks in January. By when should I be done applying for GOES and schedule the appointment with PSK? Also, do I schedule it myself or am I randomly given a date to come to PSK? I am just trying to come up with a viable timeline for the process before getting my tickets done. Looks like the Indian process takes a whole lot of time so I can plan to come back to US just a few days after the police verification.
I am just wondering if the steps at the PSK will be smooth and fast enough given the ineffective bureaucracy.

3. Does the H1B visa need to be physically stamped on the passport for either of the steps in the previous question?

4. Has anyone's application moved ahead over to CBP ever since you applied for Global Entry?

Thanks in advance!