H1B to H4 to H1B question

[dsamyu]

Hi, I had been on H1B for 6 years and went on H4 (my current status). While on H4, I had I-140 (not employer based) approved. Currently I have a University interested in sponsoring H1B ( 1 year extension under AC21 ???). My prospective employer is filing my H1B by the end of Aug 2018 under premium processing. My H4 and I-94 are due to expire early september. However, the job offer start date is early october. My questions

1. Considering it is filed timely and approved, will I be considered out of status during the gap period between early september (H4 expiry) and early october (job start date).

2. Would it be best for me to travel out of the country before they file my H1B and go through consular processing?

3. Is the H1B approval date my job start date?

I want to make a decision whether to have my employer file for H1B while I stay in the US or  to travel out, have them file my H1B and go through consular processing. 

Thank you in advance for your answers.

[pontevecchio]

You can get a 3 year extension based on an approved I-140. If they do not give you an I94 with approval, then leave and return with the visa. If it is a recognized university, they should be able to tell you the best way.