Is it permissible to travel to Western countries to complete university studies, despite the mixing and temptations involved, and the risk of a general punishment befalling them due to their corruption, knowing that the goal is to serve Islam and Muslims? And does marriage before traveling protect against these temptations?

The criterion for the permissibility of residing in lands of disbelief is being secure from tribulation and being able to establish one's religion. If you are secure from the trials of desires and doubts, and the knowledge you seek is not available in Muslim lands, then there is no harm in you traveling. What you mentioned, that they might be afflicted with the punishment of the people of Lut, is a mere conjecture that cannot be relied upon. Your marriage before traveling to those lands is better and more protective for you. However, if you are not secure from temptations and doubts, then it is not permissible for you to travel.