I don't understand why Alcatraz has to climb the alien towers. WTF are those things, what happens when Alcatraz gets in them? Is the suit fighting the virus or something?

An area denial Bio-weapon. That's what it was. :/

The spire things release both intense heat and infectious spores when activated.
When Alcatraz climbs into them the spires are disabled, and the suit is a little bit closer to developing a vaccine.

High concentrations of Alien cancer, you know the one the whole city has? The suit gets purposely infected and tries to integrate with the disease, not until Hargreaves gives him the shot does the suit create the vaccine.