I will start with your last question, as that's the most complex. You gave no indication of the size of the rescue team, how many people they are rescuing, or how much and what type of opposition they will have to fight through. Each of these could radically change the composition of the rescue team based on the answers to each variable.
Note: The US military pay grade is specified to help illustrate each person's relative position in the group. Pay grades starting with 'E' are enlisted men, while those with 'O' denote officers. Refer to http://www.army.mil/symbols/armyranks.html
for comparisons and insignia.
A rescue team could be as small as an infantry squad, made up of two "fire teams" of four or five privates (E-2 or E-3) each, equipped with an M-4 5.56mm assault rifle and led by a Corporal (E-4) or Sergeant (E-5). One M-4 in each squad would be equipped with an M203 grtenade launcher attached to it. The squad leader would be a sergeant (E-5) or Staff Sergeant (E-6).
It might be an infantry platoon, made up of three or four standard squads as described above (riflemen) and one weapons squad of four privates - two gunners and two loaders - led by a Corporal or Sergeant, that is equipped with 2 belt-fed machine guns such as an M-240 (7.62mm) or an M249 (5.56mm). The platoon is led by a second lieutenant O-1 (rarely a first lieutenant O-2) who is assisted by a Sergeant 1st Class (E-7) acting as the Platoon Sergeant.
An infantry company is made up of four to six platoons, one of which is a heavy weapons platoon equipped with machine guns described above and occasionally the .50 caliber Browning M2 12.7mm machine gun, M224 60mm light mortars, M-203 or MM-1 40mm grenade launchers, or M-79 light anti-tank rickets. An infantry company is usually commanded by a captain (O-3)/ His Headquarters Section is led by a 1st lieutenant who as company executive officer (XO) is 2nd in command, a company clerk, a Company Gunnery Sergeant (E-7) and/or First Sergeant (E-8), a supply sergeant, an armorer/assistant supply sergeant, a master gunner, a gunner, an NBC(Nuclear/Bacterialogical/Chemical Warfare) NCO (Non-Commissioned officer, or sergeant_, three drivers, and one radio telephone operator.
The unit going up in size are batallion (commanded by a major - O-4), regiment (colonel - O-6), brigade (brigadier general - O7), division (magor general - O-7 or lieutenant general - O-8), and corps (lieutenant general - O-8 or General - O-9), each consisting of three to five of the units under it. Also, as the unit level increases, more special units are attached for support, such as mortar batteries, artillary batteries, tank companies, engineer units, helicopter gunship squadrons, etc. The result is that whila company would consist of 100-200 soldiers, a division would consist of anywhere from 10,000 and 20,000 soldiers and be led by a lieutenant general (O-8) assisted by a staff of over 100 staff specialists of all ranks.
In answering your question regarding military teams and there leaders, I also tried to cover the typical weaponry involved. These all refer to United States Army infantry units; other unit types like artillary, armored, airborne, or air assault would be different in manpower and equipment to match the job. Units from other countries would be even more different in both manpower and equipment, following the military doctrine that country adopted.
This is obviously a tiny sliver of what makes up a military unit; hopefully it's enough for your story.
I think the expression you are looking for when you say "arming a gun" is "chamber a round", orthe slang "lock and "load", both of which mean placing ammunition into the gun to make it more than just an expensive club.
Last note: NO ONE in the military ever refers to their weapons as "gun" - it is ALWAYS refered to as a "weapon".