Can I apply for a work permit on a tourist visa?

No you are not allowed to apply for a work permit on a tourist visa. It would be best for you to talk to your employer's immigration attorney to...

My mother was petition by her sister in 4/2001 was approved in 2/2009 but nothing yet. Now my brother is a US citizen

Your mother may qualify for 245i. At this point the best thing for you to do is bring all of your paper work to an attorney to evaluate your options.

Can i?

It will depend on whether your J visa is subject to the requirement to be abroad for 2 years. It would be good to meet with an attorney to...

I -130 denied......?

I agree with my colleague. The fact that you are now in deportation/removal proceedings means that you need to meet with an attorney as soon as...

Canada/Australia as backup for Green card?

I agree with my colleagues. Your best course of action is to consult with a Canadian immigration attorney.

I agree with my colleague. Generally only a US consulate can schedule your fiance's interview.

I agree with my colleagues. You should be able to fly. You may want to bring a copy of the I-797C receipt notice.

Can I apply for Employment Authorization Document (EAD - form i-765) with Conditionally Approved I-360 (pending interview)?

Yes you can apply for work authorization, You should read the instructions for the I-765 to determine the correct code.

Is it safe to start K1 fiancé visa process a few days or weeks before my fiancé's possible "bar" or "inadmissibility" ends?

It is a good idea to meet with an attorney in person for a consultation to evaluate your options. If she is subject to a bar it may be for 3 years...

Im doing adjustment of status and i have a record .does uscis need certified copies of court disposition or can it beregularcopy

I agree with my colleagues. You should send a copy to USCIS with the filing of the green card application. You should bring an original certified...