Can I keep my baggage at the front?

Yes

Inside the Art Institute there's a place where you can let your bags securely, they will give you a ticket to reclaim your bags when finish enjoying your visit to the exhibitions.

Yes, just before scanning your ticket at the entrance there's a coat check, bags are not allowed in the museum.

Yeah, they have bag check and typically require you to check your bag

No but I’d try the bell hop across the street probably 20 buck but you would know you can get 24/7 and secure