Is there a shuttle bus from the train station to the mall?

This is the info I found on Tanger Outlets; "Tanger Outlets is conveniently located a short distance from the Deer Park LIRR Train Station. Purchase the Tanger Deer Park package at open ticket offices and ticket machines (gray or blue). Package includes rail fare to Deer Park station and a short shuttle ride to Tanger Outlets. Turn in the voucher stub at Tanger Outlets Shoppers Services for a coupon book worth hundreds in savings. Click the link below for Tanger package and train information or call 718-217-LIRR."

Yes there is.

Yes

Yes, there is. The shuttle bus picks up on the north side of the tracks.

🙃

Yes there is a shuttle bus from the Deer Park train station to the outlet

There is a bus stop that goes inside the mall I'm not sure of the bus but there is one

There was, I'm not sure if it's still available. You might want to check the website for Tangers outlets or the LIRR website.