The infantry platoon in Vietnam was a culmination of experiences from the WWII and immediate post-war era, as you suggested. It took many concepts that were either in their infancy, or deeply flawed in execution, and attempted to perfect them.

The fireteam system was probably the most drastic change for US Army squads; a concept that was in use in a form by US Marines by war's end, the concept of permanent fire teams in a squad hadn't become norm even by Korea. In WWII, US Rifle squads were large, often 10 men or more, and split into 'elements'; pseduo-fireteams that could change based on the Squad Leader's whim but were generally divided into a lead, trail (often containing the BAR) and scout elements. As the war progressed, it was deemed ineffectual for every squad to have its own scout elements and the practice was to operate in two elements instead. This evolved into a smaller squad during the Korean war with no elements. The idea behind this was to ease the burden of command on a junior leader. By the end of both WWII and the Korean conflict, it wasn't uncommon to see squads carrying multiple automatics, as the semi-auto battle rifles proved greatly useful, but not sufficient on their own to establish fire superiority.

MGs per squad

Vietnam put all this into official practice. Combining the lessons of Korea and WWII and resulting in a: 10 man squad consisting of 2 fire teams of identical make-up. Both fireteams would have an automatic rifleman, rifleman and a grenadier. This same basic make-up is still in use today with the US Military and is in line with having all fireteams conduct lead ("Manouver") and trail ("Fire") with equal competency.

cooperation with other branches

Vietnam wasn't exactly an area ripe with opportunities for Combined Arms, nor was Korea. I'll extrapolate to the period itself. In short; the US Army's combined arms efforts were almost entirely based off their end-game conclusions in the immediate years following WWII. The "Cavalry" and "Armored" Branches were combined in the 50s to avoid needless branches, and Armor was generally seen to be viewed as a tool for shock-action or a weapon for maneuver. This ended more or less decisively the debate that had begun in the 1930s as to who was to be responsible for the command and development of armor; the Infantry as supporting arms, the Artillery as 'flying guns' or the Cavalry as a technological successor.

The basis of the US Armored division in WWII led rise to the Brigade Combat teams that became standard post-Vietnam for the US military. Flexible or semi-permanent brigades became permanent, combined arms brigades that made up a division. This same basic structure is in use today with the US military with their Infantry, Stryker and Armored Brigade Combat Teams.

Combined Arms in short therefore, permeated almost every tactical formation of the US Army at the time. An example then; the 25th Infantry Division at one point in Vietnam consisted of 3 Brigades; lets look at its 2nd Brigade:

-1st Battalion (Mechanized), 5th Infantry

-1st Battalion, 27th Infantry

-2nd Battalion, 27th Infantry

-2 Battalion, 8th Artillery

Every brigade in this formation echoed this formation; creating a flexible 'infantry' formation that, if coupled with Corps-level tank units could easily conduct maneuver warfare in conjunction with the mechanized battalions in each brigade.

Source(s):

US Army Infantryman in Vietnam 1965-73 (Rottmann, Gordon L.)

Through Mobility We Conquer: The Mechanization of the US Cavalry (Hofmann, George F.)

The Organization of the Marine Infantry Regiment - 1944; read it free here