Sport mode are meant to get slightly better acceleration when floored by changing the transmission's shift points to hit max rpm and/or increase throttle sensitivity, so having sport mode on while normal driving doesn't serve any purpose but it will not cause any damage.

Your engine is designed to last so many billions of revolutions before it wears out. Engines have a certain degree of tolerance, once you exceed those tolerances due to wear and tear, you will start burning more oil, etc. If you floor it constantly, you will wear it out faster. If you drive like a sane person and go at 2k RPMs, it won't hurt anything. It takes the same amount of fuel to go at 2k RPMs, no matter if you have it in Sport Mode or not. 

Yes but your mpg may decrease 

I use sports mode when I need to accelerate really fast. Like getting onto a freeway. Or overtaking other cars on a freeway. 

Otherwise I am in Normal mode. 

I tried Eco mode, and it is ok. I much prefer better throttle response. Eco mode just lags. There is a time and a place for it, if you know are just cruising and don’t need to accelerate fast. 

Posted by: @chaseboy4

is it okay to drive a Grand Cherokee in Sport Mode on regular basis

Yes. It doesn't hurt the car, might sting your wallet a bit..

Posted by: @chaseboy4

What happens if I were to go over the green flag icon?

I assume that's something on your instrument panel. What does the owner's manual say?