The Great Lakes are in the northern part of the US, and Texas is in the Southwest part of the US. The Great Lakes are therefore not near Texas; however, the Mississippi River borders Texas and empties into the Gulf of Mexico.
No. The US states that border the country of Mexico are California, Arizona, New Mexico and Texas. If you were asking about the US state of New Mexico, Minnesota doesn't border such state, either. Minnesota borders Canada, both Dakotas, Iowa and Wisconsin.