Will part time H1b( 20-30 hrs/week) have any problem for stamping for first time ?

If it is the first time, it should be in your home country or where you studied.

Im 31 yrs old. My mom is a greencard holder. Iam overstayed in the us. Is my mom can still petition me?

Your mom can petition, but you may not be able to adjust status.

My April 2013 petition for H1B was approved. Are there any dangers to waiting to get my visa stamped?

Once an H-1B in your behalf is approved you are cap exempt unless you do not use it for next six years.

Filing future labor from same employer for different position in EB2 earlier filed in EB3 Category

Work with your employer and their immigration attorney. PERM representation is usually joint and so the attorney should be able to answer most of...

H1B Transfer for same period

Yes

I changed from F2 to F1 status in USA. I need stamping of F1 in my country. How risky is this?

As long as you continue to have strong ties to your home country, you should be fine.

Can I travel abroad since I have pending I - 485 and pending I-131?

No, not without an approved advance parole document.

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Is it possible for B2 visa holder to adjust a status when sponsored by employer?

The process is longer than 5 months. Alien is unlikely to adjust status in such a short time.

In filing alien petition I-130, after your biometrics are being taking, what is the next step

Do you mean I-130 with I-485?

It is necessary to change I-20 to Cap Gap status during cap gap period?

Take the receipt/ approval notice to your DSO.