Is it ok to shoot Hornady CRITICAL DEFENSE 110gr ammo in a SMITH & WESSON Model 37 .38 alloy frame revolver or any light weight revolver and what is the ups and downs of the light weight round ?

Perfectly safe. The 110 gr. bullet was developed a long time ago for use in J-framed size guns where larger K-frames would handle the
38 +p ammo. The 110 traveled faster then 158gr and expanded well.

Hornady just put the flex-tip on an old design of a hollow point 110gr.

About twenty years ago I carried 110 gr. bullets in my Model 60 until I went with an auto loader.

The standard 110gr Jhp in a snub nose runs around 875 fps with 187 ft/lbs of energy and some gel test show around 9 inches of penetration but these are just ball park figures.

Nothing wrong with standard ammo but you are working with the minimum energy in the revolver with what is called a SD type load.
Some like the other route, of a BIG SLOW bullet like the 158 Lswc target load at around 680 fps for a non +p load energy of 162 ft/lbs. At 730 fps it will match the 110gr in energy but one needs a chrony to find out what fps his guns are getting.

One note, the lighter bullet will have LESS recoil but it might also not give as much PENETRATION as the heavier bullet.
If you can place shots accurately, I see no problems with the lighter bullets , if the BG is not wearing a Pea Coat and three layers of clothing.

The gun will certainly handle the round, but I think there are better choices. The Speer Gold Dot 135gr. +P is getting good reviews and approval by some major PD's. The old FBI load, +P 158gr. LSWCHP, has worked well for decades; and now Buffalo Bore is producing a standard-pressure version that appears to duplicate or even exceed the velocity and energy of the +P.

The 110gr. will get the job done. I'm just a fan of heavier bullets. Old, you know...

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Oh well, what the hell.