VIDEO of the car trying to start below....

Drove my moms car today (1988 300SEL) because my work car is still in the shop.

Anyway I drove the car to work, then to my grandmothers after, then to the supermarket after that, it ran perfectly. Came out of the market and drove across the parking lot to get gas and I filled it up... then I went to start it and it only cranks and will not start! Toward the "end" of when I am turning the engine over, it will occasionally and very momentarily sputter and stall.

I am guessing fuel pump, fuel pump relay or? I have no idea. I know I have owned the car nearly 3 years (except the past 3 months she has had it) and its never gave me or her a problem until now. We had to push it away from the pumps and to a parking space in the store lot until we can try to figure out where to start or a shop to take it to, but trying to get some ideas. Oh, also the check engine light comes on momentarily when first started after you have been driving for a minute and goes off after 2 minutes, but not sure if the problem is related or not?

Thanks in advance for any help!