I am planning to shoot some scenes for a student film here, is there some form I need to fill out? Or is there someone I need to ask permission from? Or is it free game?

No, you may go and shoot all the film here yyou need, but yes this is a dog park area, and thereis people and dogs about, so be aware of this

I think it depends.if this is something official that you wil need to close a part of the park then I'd say yes contact the city but if this is an informal shooting and you may be inadvertently interrupted by dogd and people yhen i say it is free to go