What that higher form of war means, is even unclear to "Steve". It seems it is meant to be unclear, and maybe "Thor" will get a chance to explain it later on, since "Steve" asks the same question as you.
A higher form usually means a more advanced, nobler, or more complicated version of something. Which, in the case of war could mean a lot of things, depending on who says it.
As for the unfinished sentence, without further context, it seems likely that "Steve" would be neck-deep into anything that he would be into if he were still making weapons. I guess it would be warfare, possibly making money out of the war that they seem to be in. Ask yourself what whoever is making the weapons for this war has a lot of, and that is what Steve would have had a lot of as well :)
The last one is the simplest question, as the answer is right there in the script:
TONY : [..]How is this now about me?
STEVE : I'm sorry, isn't everything (about you)?
Examples occur whenever someone leaves out a verb, a noun or a phrase in order not to repeat it:
The oldest sister is beautiful!
Aren't they all (beautiful)?
At X is used if X is a place or point. If there is a list of something, especially if it's sequenced in numerical or alphabetical order, a single item of that list is considered a "point". So phone numbers would take at as the phone company has a big list of numbers to give out and yours exists on that list.
Call on X is a phrasal verb meaning to use X as a resource or ask X for help.
Call up X would emphasize the standard meaning of call without phrasal modification.
Call X on Y means to place a call to X using Y; Y would be a type of communications equipment or software, never a phone number. You could say Call X on Y at Z where Z is the phone number.
Best Answer
The first instance represents a shortened form of "call the play". This term derives from American Football where it refers to when a team is told what they're going to (attempt to) do after the next restart - the person who decides this is usually either the coach or the quarterback. Colloquially, it can refer to taking ownership of a decision and making it. In the context, "it" is "the play", ie the strategy that the Avengers will use to deal with the situation.
Your interpretation of the second instance is correct - it derives from the formal requirement of medical professionals to make a note of the time of death for patients they've been treating - more fully: "they've called the time of death".
As per msam's answer, "down" doesn't necessarily mean dead, it could just mean "out of operation", it's commonly applied to people who have been shot and are incapacitated.