arm and elbow position -- sticking out or straight, also pushing or pulling on handlebar
Ultimately all road bike positions are a compromise between comfort, power and aerodynamics.  The balance between each component depends on your goals, experience, flexibility and any underlying injuries or physical dysfunctions.  If you find a position that works for you, that is outside the "typical" road positioning, then you should consider it as valid as more typical positions. A good fitter should be able to accommodate unique personal characteristics. That said, you must also be careful in that non-stand positions may also lead to other unintended problems over time if you do not have a good awareness of how your body operates biomechanically.

What follows are some answers to some of your explicit in-text questions.

Elbows in or out


  The fitter told me I should not stick my elbows out (which I do). He said my arms should be a little flexed at the elbows but in a straight line when viewed from the top (ie not sticking out). 


A lot of road positioning is about reducing frontal area. Keeping your elbows in reduces your frontal area thereby reducing the amount of new turbulence created as you move through the air.  Reducing your frontal area allows for greater speeds to be attained for the same level of effort.  

While critical for racing, a recreational rider may benefit from comfort over aerodynamics.  I recently shifted to much wider road bars (44 cm instead of 42) with an additional flare in the drops (which adds another 4 cm).  This sticks my elbow out relative to more standard road bars.  I know I am less aerodynamic with this setup, but I am also a lot more comfortable.  It opens up my breathing, and makes the bike easier to handle on mixed terrain (i.e., gravel and dirt).  For me this compromise is worth the loss of efficiency (for this bike).

Pulling or pushing on the hoods


  He also said I should be gripping the hoods with my hands pretty strongly and pulling rather than pushing on them, which uses abdominal muscles more than shoulders. 


I am suspicious your fitter is really only considering performance fits.  If you are racing or doing large efforts (e.g., 250+ watts sustained effort) you need a way to resist the force generated by your legs.  In this case a low body position and pulling back on the hoods works well.  If you are riding in a more relaxed pace (e.g., under 175 watts), this body position makes little sense as you have less force to resist.  This is why touring and endurance positions typically have shorter reach and more upright (higher stack) as you do not need as closed of a body angle to resist the force generated by your legs.  

If your cockpit is setup too long and low for the type of riding you do (e.g., performance fit cockpit given to an endurance or recreational rider), then you will feel you need to consistently push against the hoods to keep yourself upright as you are not generating enough force from your legs to fully support your upper body weight with your core muscles (i.e., your core muscle do not have enough force to resist against to keep you upright so you rely on your arms).

Neck Pain


  He said that neck pain can be caused by pushing on the bars which hunches my shoulders up.


While it is true that neck and back pain can be caused by hunching up the shoulders there are multiple possible sources that can cause you to hunch your shoulders (not just the one pushing on the bars).  For example, a cockpit that has a reach that is too short and a stack that is too low can also cause you to hunch your shoulders.  Even how your pelvis is rotated on the saddle can affect your spinal alignment leading to hunching and neck and back pain (personal experience).  

Ultimately, fit is a tricky subject, often depends on an individuals goals, underlying flexibility and pre-existing injuries.  It is also a bit of a moving target that can change day to day especially if you have unresolved injuries or flexibility issues.