Will I be able to fix my fiance's papers?

Your question requires an attorney consultation. It is not a simple question that can be answered on this type of forum. There are many factors that would need to be considered and evaluated. I strongly suggest that you contact an experienced immigration attorney for a face-to-face consultation and give him/her all of the facts surrounding your situation. (S)he would then be in a better position to analyze your case, advise you of your options and, if possible, make sure that the necessary paperwork and documentation is properly presented so as not to delay the process.

If you are either a legal resident LPR or a U.S. citizen, you may be able to fix your fiances immigration status. How you can do this depends on various factors including but not limited to his physical location for instance. If he is presently in Mexico, you may be able to first file a form I-129F fiance petition which, if approved, he can come to the U.S. temporarily and thereafter you must marry him within 90 days and file for his permanent papers green card. This option is subject to several requirements and is only availing if you are a U.S. citizen. If physically present in the U.S., fiance petition is not an option and instead you must first marry and then file a form I-130 petition with CIS. This second option also has several requirements and although, marriage is an essential requirement, it does not guarantee a visa will in fact be issued. There are several issues to consider under either option and they cannot be fully discussed in this forum. Instead, you should must consult with an experience immigration attorney.

The chances of success are normally good, but it depends on how he entered the country, has he left and returned, does he have deportations or voluntary returns, etc. You should schedule a consultation with an attorney as it can get quite complex and you need to provide more detail.

It would depend on all of the facts. A USA citizen can sponsor a fiance abroad to get a K-1 fiance visa.