Can foreigners become USA residents by purchasing a home in the USA?

Buying a home in the US is not going to get immigration status for your parents.

Yes you won and may proceed with filing for the green card.

How do I answer the "immigration proceedings" question after withdrawing my application for admission?

Given your stated facts you were not in immigration proceedings.

Yes but given they are traffic infractions they will not have any weight.

Section 13 of INA (1957) is used by diplomats To obtain a green card.

Will I get a chance to have a legal status again?

You may be able to get approval again but you will need to ask the court for voluntary departure, reapply from home country, present a great...

Can i file the application without the taxpayer or sponsor? if not what should i do?

There is no substitution for a qualified sponsor.

Any time one deals with immigration it is my opinion they should take advantage of legal representation. Having said that, if you have put together...

What is the filing fee for the initial I-765 when I file it with I-485 concurrently under the (c)(9) category?

There is no fee required when filing with I485.

How should I divorce in Texas if our marriage certificate is from another country?

To file in Texas you must be a resident of the state for 6 months and the county where you will file for 3 months. Once divorced (at least 60+ days...