Extremely bad. Huge areas of Mexico are essentially war zones and the police has no control over it.Last year 14,771 were killed in the cartel battles, thats insane.To give you some context more people died in Mexico than Iraq(13,187) and Afghanistan(13,997)

This BBC article is pretty good so I suggest you read it if you want to know more.

They are completely barbaric,from dissolving students in acids to decapitating politicians and rivals.You name,they did it.

How could a government stay functional if it loses the monopoly of violence within its national border?

It doesn't. Many areas are lost to cartels completely and they have to routinely bring their own military to deal with the huge problems.

How do the criminal activities affect the political environment of Mexico?

I am not Mexican so I am not completely sure, but I don't think they have a particularly healthy political environment.Like you said, 1/4 presidential candidates were murdered,Mayors get beaten to death , Decapitated or worse if they don't listen to the cartels.

If you look at Mexico, it should be a paradise. It has two oceans on both sides,connected to the largest economy on Earth and also NAFTA.They can speak spanish and are connected to the south america,have a large and young labor force, have decent education.It has massive potential but is ruined largely due to the cartels.