There used to be parking right next to it that was reserved for the restaurant, but the local alderman killed that out of spite when he himself couldn't find parking up close without going into the beach spot. That's self-absorbed ignorance on his part, but you can still find, it's just parking is hit or miss.
Yes. You may have to pay, but it's there and not too far.
Yes. You may have to park on the west side of LSD and walk the underpass if it's busy
Yes, there is parking.
Parking is available if you get there early enough. Gets crowded on a nice day
Thanks! Your answer is awaiting moderation.