Also, would they put people from the same place in the same unit, like the Biritsh did during the first part of WWI?
Your geographic location could or could not be a factor in where you fought depending upon which branch of service or type of unit you were in. For the Marine Corps and Navy, your prewar location did not really matter; however the Navy had a (not very strict) practice of separating siblings, so as not to have them serving on the same ship. After the five Sullivan brothers were all killed when the USS Juneau sank, this policy was presumably enforced far more strictly.
The Army was a special case, however:
The Regular Army (those divisions that existed prior to the US entry into WWII), the Organized Reserves (inactive divisions that existed only "on paper"), and the Army of the United States (new divisions made up men who were drafted) took men from all over the country and placed them in any division. In this case, where you were from really had no bearing on your final theater of combat.
Army divisions existing prior to the US entry into WWII:
Organized Reserve divisions (most served in World War I and were deactivated after the war; reactivated for service in WWII)
Army of the United States Divisions (these divisions, save the 42nd and 43rd, were completely new and had no previous lineage)
National Guard divisions of WWII originally consisted of men from certain areas of the country. In this case, where you resided did have an effect on your theater of combat. For example, if you lived in Wisconsin and were part of the National Guard, you would go to the Pacific as part of the 32nd Infantry Division. If you lived in Minnesota and were again part of the Guard, you would go to North Africa as part of the 34th Infantry Division, and so on. However, "draftee" regiments of the Army of the United States were used to bulk up some divisions to prepare them for war; The 35th Infantry Division consisted of the 134th Infantry Regiment from Nebraska, the 137th Infantry Regiment from Missouri, and the 320th Infantry Regiment that was constituted from draftees.
National Guard Divisions (the recruiting patterns for those Divisions that fought in World War I were mostly the same in WWII)
Two unique cases:
As the war dragged on and casualties mounted, men from all over the United States were used to fill spots in the National Guard divisions. Due to this, by the end of the war you could not tell a National Guard regiment was originally from Texas or Mississippi.
In relation to your second point, men from the same area were sometimes placed in the same units, sometimes not; you would have to ask around a bit to find someone from the same state as you in a Regular Army, reserves, or Army of the United States division. The National Guard was different; whole companies or even regiments came from one city. In more sparsely populated states, the men of a few counties would form a company, with the armory in the nearest large town.
This could have tragic consequences, and situations like the "Pals Battalions" of the First World War did occur. I give several examples below. Here are the 29th Infantry Division's armories in 1941:
Headquarters, 29th Infantry Division- Fort Meade, MD
Headquarters, 115th Infantry Regiment- Frederick, MD
Headquarters, 116th Infantry Regiment- Roanoke, VA
Headquarters, 175th Infantry Regiment- Baltimore, MD
Bedford, Virginia, the home of the armory of Company A of the 116th Infantry Regiment, paid the price; nineteen men from the town of about 4,000 residents were killed on D-Day. Nearly 100 out of the 155 men of Company A that landed on Omaha Beach did not survive. Less than a dozen escaped unwounded.
Another particularly tragic example is the fate of the entirety of southwest Iowa after the battle of the Kasserine Pass. The 168th Infantry Regiment was basically destroyed, with over 1,800 men being taken prisoner. An excerpt from Rick Atkinson's haunting subchapter on the subject;
"Wild rumors flourished. Shenandoah, Iowa, supposedly had lost 500 men -- even though barely one-quarter that number were serving in North Africa. The truth was grim enough. Clarinda had lost forty-one, Atlantic forty-six, Glenwood thirty-nine, Council Bluffs thirty-six, Shenandoah twenty-three, Villisca nine. Red Oak's total reached forty-five, nearly a third of Company M, which altogether had lost 153 men, among them the commander and six lieutenants. Total losses for the 168th Infantry Regiment included 109 officers and 1,797 enlisted men."
Sources:
http://www.history.army.mil/html/forcestruc/cbtchron/infdiv.html
An Army at Dawn: The War in North Africa 1942-1943, by Rick Atkinson
Beyond the Beachhead: The 29th Infantry Division in Normandy, by Joseph Balkoski
Omaha Beach, by Joseph Balkoski
For the most part, no. Many of the main U.S. Army and Marine divisions (along with the Navy) intermixed men from all over the country. Many of the U.S. Army's divisions, Marine units, and Navy ship crews contained men from all states, territories, and Indian Reservations. Consequently, a person's home rarely influenced the place he was sent to fight. There were a few exceptions to this procedure.
After several months, US Army National Guard units began to take part in campaigning and battles. Many of these divisions were composed of men from the same state. One example that comes to mind, is the 29th infantry division which was a composite of three earlier Virginian infantry and artillery regiments. Interestingly, many of the Americans who died on on D-Day were Virginians due to the fact that 116th Regiment, part of the 29th infantry division, landed at Omaha beach.
There were also segregated military units which restricted the places its members could go while allowing many from similar homes to serve together. Japanese Americans from Hawaii and Western internment camps served in the 442nd Infantry Regiment (the most decorated American military unit in history). Because of their Japanese ancestry, they were prohibited from fighting in the Pacific and instead sent to the European theatre where they distinguished themselves in Italy, France, and Germany. Additionally, the 65th Infantry Regiment, sometimes called the Borinqueneers, was an Puerto Rican unit which served in multiple conflicts, including WWII.
Sometimes, a person's unique skill set could influence the unit and locations he was placed in. Navajo Indians from Reservations in Arizona and New Mexico, were drafted and served as code talkers in Marine divisions. Their unique and little known language made them a valuable asset, notably in the Battle for Saipan.
In addition to a person's language skills, a persons local knowledge could designate them for a specific unit and theater of war. After the Japanese invade the Aleutian Islands, the American military formed the 1st Alaskan Combat Intelligence Platoon, other wise known as Castner's Cutthroats, which consisted of local Alaskans familiar with the terrain, geography, and weather of the territory. Their experience and know how contributed to American victories at Attu and Kiska.
Some Units' mission statement and purpose sometimes attracted men from the same area. The 10th Mountain Division began and modeled itself after the German and Finnish snow-ski units. As a result, the division, a battalion at the time, hired and recruited top college skiers and athletes. A core group came from New England states like New York, New Hampshire, and Vermont with an addition of a several westerners.
Finally, the military branch of the soldier could determine his location. Unlike the Navy, Army, and Coast Guard, the United State Marine Corps (excluding small detachments in Europe) only served in the Pacific. Thus, if you signed up with the Marines, you would be fighting the Japanese in the island hopping campaigns.