Am I subject to the 2 years rule?

Request an advisory opinion from DOS. That's the most trustworthy authority on this.

H1B check not cashed, no receipt?

Yes, that is a very safe assumption, given that we are already in July.

Consult with an experienced lawyer in private and full confidence to learn about the options that may be available to you.

Can I apply to financial aid if I have conditional residence?

Conditional residency or not, as an LPR you are still eligible for financial aid for academic studies.

Yes, simply appear at the ASC without an appointment letter and you will be taken out of turn, given that need to travel now. Don't worry. Routine.

Can Canadian citizen travel to the US while I-824 is pending?

Yes, as a Canadian visitor on a visa waiver and remain for up to 6 months.

I94 Admit until Date later than the admit until date stamped on passport, Difference is only of 10 days. Which date to follow?

In such situations it is always the I-94 admit until date which controls/defines the terms of one's stay in the USA.

How can I prove paternity without a birth certificate signature for an Italian citizenship?

Not a matter concerning US immigration law.

Can I apply green card for my parents which they are living in Iran?

You definitely still can, however, it will take a long time for them to get approved and issued immigrant visas, due to the travel ban in effect...

Can i travel with an expired I-94 from AUSTIN TO SANDIEGO with a extension letter which says case under process

Yes, on your valid passport. There is no need to produce or show anything else.