There are pavilions with tables. But I'm not sure about bringing a table of your own. Try calling Spanish Fork City and ask for recreation for help.
I have taken a fold out table for my family picnics, as well as folks out chairs. It felt natural and nobody said anything.
I've seen people set them up all the time. Should be fine.
I don't think so but there are sitting areas in the park.
Thanks! Your answer is awaiting moderation.