Yes, this is done all the time. Best to wait 60 days after the entry to marry so it doesn't look like you made a misrepresentation. File the relative petition and adjustment, work authorization, and Advance Parole travel document -- best not to travel on the visitor visa after you've married. You might end up stuck outside the U.S.
You can legally bring your fiancee from Mexico with a "fiancee Visa" and you will have 3 months to marry him.
No. Not even marriage to the father grants gives you citizenship. You have to apply for naturalization.
It depends on the exact circumstances and the residency status of the various people involved. For example, if a US citizen marries a foreigner, they don't automatically lose their US citizenship. If that foreigner is later deported, the citizen is legally entitled to remain in the US. However, if the wife and kids are not US citizens but entered as dependents of a permanent resident who was later deported, the answer would be different (again: depending on exactly what their residency status is).
Only if they're in the US LEGALLY!
You're a resident of the country in which you're currently living, as long as you're doing so legally (as a citizen of the country or as a foreigner not in violation of visa requirements - tourism visa don't count)
No, your fiancee would need to be a citizen to petition you. Moreover, if you entered the US illegally, then even if your fiancee was a citizen, you would NOT be able to legalize your situation. If fiancee was a citizen you would need to file a I-130 petition and file for a I-601 waiver at the US embassy of your home country, and you would have to go back to your country for the hearing. Such waivers are hard to get. Your citizen child will not help you. Suppose you entered legally and overstayed because you found out you were pregnant. The situation is different in this case. You would be able to extend yopur stay, get a visa initially, then a Green Card and then citizenship after 5 years in Green Card Status
Marriage is a wonderful event. Yes, if a female US citizen legally marries a non US citizen in the state of Michigan, they are indeed legally married.
If your willing to Fack the local priest than there is no problem.
If you travel to the US on a k1 fiancee visa, you generally have to apply for a temporary work permit once you arrive. This may take a few months to process, so under normal circumstances, I would say that no, you can't legally work as soon as you arrive.
Who is the legally recognized father? The U.S. citizen? If not, it may be advisable to get documentation indicating she is legally permitted to have the child immigrate to U.S. An immigration attorney in the Philippines familiar with these particular circumstances is a good choice in this situation. http://www.dixonimmigration.com/index.php?pid=2
It depends on whether or not you are here legally. You will not legally be a citizen until you have lived in the US for four years after you have bought a home in the US.
Not legally.