Working with i-797 receipt notice based on TPS?

No, you can not start working until you have EAD card with you

How long would it take for our I-129F to be reviewed?

it depends, some petitions come early and some takes long time. You can visit on B-2 visa, if CBP finds out at the airport about I-130 filed and...

Cost for green card thru H1B

your employer should be taking care of these fees you have mentioned at least up to I-140 process. Anyway each attorney has different legal fees...

Does civil engineer need a license to perform the job you have applied for, If yes, then you need to clarify it. Discuss it with your attorney.

I-94 Expiration Date

Admit until date shows that you are legally authorized to remain in the US until October 10, 2015.

H4 to L1B stamping suggestions ?.

If you have changed your status to H-1B, you can check with U.S consulate in Mexico and Canada, but normally, first H-1B stamp will be done by U.S...

I am on L-1B visa, am I allowed to go in and out the US without visiting the US embassy in my home country?

it depends on the visa stamp in your passport. If it is valid you can travel back and forth until its validity date.

USCIS update

Interview and fingerprinting will be for beneficiary, not the applicant who filed the application and it is processed in consulate of your country.

Switching employers while waiting on H1B with approved I-140. Good or bad? What to consider?

Yes, you can retain your priority date, but new employer will have to file new labor cert and I-140.

Concurrently filed i130 and i485 based on spouse already in the US overstayed was denied in 2014. Now i130 approved in 04/2015

If you don't have an Attorney, then you should retain attorney immediately for your case. it seems, you have issues, can not be discussed on this...