I just purchased a Sport 2 with Magpul MBUS rear sight and A2 front sight. I am thinking about either a Burris or Vortex red dot or a Eurotech sight although they are pricey. Do I need to remove the A2 post before mounting any of these sights?

Short answer... No. You will need to learn about co-witness though.

I guess that means I have to decide whether to rely on always having batteries for the sight or have the back up capability of hard sights. It appears from the video that you do lose some of your target acquisition capability if you have either absolute or 1/3 lower co witness. Is that correct?

To make the most of using a RDS you should shoot with both eyes open. If you already train or are willing to train to shoot with both eyes open when employing the iron sights you will mitigate much of the issue.

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642-1, M&P15 TS

This is correct... when shooting with a red dot sight, you keep both eyes open and focus on the target down range. With practice, you won't even notice the front sight.

Not really. 1/3 lower puts that front sight post way lower in your optic picture when not using it. Remember, when a red dot is zero'd there is no lining up sights. Put the sight up to your eye (eyes both open) put the dot on target no matter where it appears in the optic and shoot. The beauty of a red dot.

For me, lower 1/3 is a perfect solution for red dot with back up irons if ever needed.

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Still carrying my S&W 642

OP...I made a suggestion on another thread..
Recommend Me A Sight.
I posted some comments and pics.
Hope it's of help and you find of interest.

Bushnell TRS-25 is a great low-cost alternative. Available for less than $100 with a riser (look for the Hi-Rise model) that will co-witness with the stock sights.

My only concern is I am left eye dominant and due to an injury had to teach myself to shoot my bow right handed. I have to close my left eye when shooting my bow.
I will shoot left handed with the AR. Will keeping both eyes open cause any problems?

There shouldn't be a problem if you keep both eyes open, no matter which shoulder you shoot from when using a red dot.