Are you allowed to put up a small gazebo for extra shade while having a family gathering at this park?

I am not sure that you can. Nonetheless, I have never seen anyone patrolling the park.

You can but it's not that big of an area so get there early to find your spot...have fun & stay safe

You can set up your own awning/tent thing.

I believe you can

There are already a few picnic tables beneath awnings, but I'm not sure if you can put up your own.

Yes