You're the one who is turned around. The M16A3 has the fixed carry handle of the M16A2 and the firecontrol group of the M16A1, which is safe-semi-full.

 

So you're also wrong that the M16A4 is the only one that goes full-auto. The M16 and the M16A1 both had safe-semi-full firecontrol groups.

The M4A1 is indeed the Army's standard battle rifle for whoever said it wasn't. 

Non-infantry units get issued M16's.  And they're not expected to use them.  Infantry gets the M4A1 w/ M68 Sight.  No one fires "from the hip"  when entering a room your weapon is at the extreme high ready, meaning it would take you .25 seconds to move that M68 scope into your eyesight.  Not to mention your wearing IBE's and if you're the pointman you're crouched when entering a room with a Kevlar on.  You may take a hit, but it'll only throw you on your ass, unless they hit your leg then you'll have a wound.  The M68 is a quick reflex scope, meaning you don't have to align anything.  You pull it up and you see that red dot and the bullet goes where it is, no matter where your eye is in relation to that sight, as long as you see the red dot it's fine.  Fully automatic is quite inappropriate for this situation as there should be 3 people behind you as well, and 3/4 will have a good shot on the enemy who appears, the 2nd guy, has an arm on the point man to pull him away if he does get shot so he has only 1 hand on the weapon he's using.  So the 2nd guy would be suppressive fire.  The M4A1's effective range with iron sights is 500m, with an optic it can be increased to roughly 800m.  And still able to hit dirt at about 1300m if I recall correctly from the IET handbook.  However it may also be 3600m.  Yes the M16A4 is more accurate, but good luck noticing it.  The M4A1 is shorter, lighter, and SLIGHTY more inaccurate than the M16A4 making it ideal for room breaching and CQB

The USMC chose the M16a4 over the US Army's Choice of the M4a1 as the primary battle rifle for the following reasons:
1. The M4 is bit short to use with a bayonet
2. Heat problems in desert conditions (during prolonged use)
3. M4 offers less precise aiming due to the four inches less sighting radius when compared with an 16A2/A3/A4.

One wonders why go with an M16a4 when you can have an M16a3. the M16a3 has the ablity to full auto unlike the M16a4 which has the triple burst firing option of the M16a2. Going full auto is something that would very useful in a CQB style engagement which happens very often in MOUT situations.

Is the reason why the USMC selected the M16a4 instead of the M16a3 because they have big attachment to burst fire rather than full auto or is there something else?

First of all, the M-16 is a piece of shit.  It has problems with heat transfer to the bolt and corrosion which makes it prone to stop.  It is also a very finely tuned piece of shit, meaning that it requires ALOT of preventive maintenance.  The A4 is probably much better than the A1, which had problems with the magazine.  Only 18 rounds could be loaded into a 20 round mag if you didnt want to experience a stoppage. (There were also problems with the powder in the ball rounds issued at the time.)  One thing I will say about the -16 is that the 5.56mm round, which is essentially a .22 on steriods, has deadly effects.  The round creates a cavity on impact doing substantial internal damage and the round breaks up into three pieces that float.  It was originally designed by Eugene Stoner of ARMalite and Colt later purchased the patent.  The -16 was jokingly referred to as "the Matell toy" because it resembled a toy rifle produced by the Matell Company in the 1960's.  As for the M-4 Carbine... I talked to an E-6 I know in the ANG a few years ago. I was told that it suffered over 2000 stoppages in '05-'06 along with over 1000 catastrophic failures.  The Army really needs to replace those weapons with something more reliable. 

As for CQB, you want an HK  MP-5, Ingram Mac-10, or a combat shotgun with a reliable sidearm as a back -up weapon.  Oh... and by the way, the M-4A1  IS NOT  standard Army issue.  That weapon is issued strictly to Army SOF's.  You obviously dont know your rifles...