1) Did the US armed forces shrink after the Cold War
Yes, the US Army went from 18 active duty divisions in 1991 to 10 a few years later. The National Guard and Army Reserve picked up some missions but they also reduced in size overall. The same happenned in the other services.
2) Are all members of the armed services qualified to go into combat?
Define "go into combat" All members of the armed services qualify in basic rifle marksmanship and basic combat skills. Depending on your specialty you receive additional traininig. Now an Infantryman's job is to close with the enemy and destroy him by fire and maneuver. Tankers focus shock action, firepower and protection into a direct fight. The artillery provides cannon support. Combat engineers operate under fire to clear and breach obstacles. But all of these folks are one big parade without the supply, maintenance, signal, transportation, MP, etc. support personnel. And you will find some of these support personnel on the edge of the battlefield recovering damaged vehicles while under fire, delivering ammunition, food and water and operating convoys. That is why the supply truck, maintenance vehicle and mess truck all have machine guns mounted on them....and they have been used, a lot. These same folks are also expected to provide a certain level of self defense when attacked. This true in the Army and Marines (the Marines like to say every Marine is a rifleman and they take it seriously).
In the Air Force it is a little different. Support personnel work to get the aircraft into the air and keep them mission ready. So are they "in the fight?" Damn straight. They keep them flying.
In the Navy and Coast Guard, EVERYONE on board a vessel is in the fight. Doesn't matter if you are quartermaster at the wheel, a radarman in the combat information center, a hull technician in the engineroom, a mess attendant in the galley or a gunners mate on a gun mount, everybody is in the fight. Because you all share the same fate.
3) Is a deployment of thousands of troops considered a "minor deployment" (this is in regards to the planned US troop deployment to Israel) - I said it wasn't , please correct me if I'm wrong.
1,000 is minor for the Army; its very big for the Coast Guard. But if I put those 1,000 sioldiers in the Antarctic and have to sustain them for a year then that ramps up th elevel of complexity. So it is a relative term and we tend not to use terms like that. As for your specific question, I do not know what deployment your are speaking about. If it is about the Patriot ABM then a 1,000 troops is about a battalion with all of its support structure. Not very big but a lot of capability.
4) Is the US military stretched thin at the moment?
Not as bad as it was in 2007 & 2008 but it is still stretched.