Do Non hearing motions exists in California?

Get a good idea of where you want to start, gun for it, work through the motions and ask for help along the way.

However be aware these types of people usually file motions and to continue to exploit legal loop holes, in order to stay in the property longer, so take no prisoners. 

At least until the tenant is out.But if you have nothing better to do and want to go through the motions of a wholesale.  

You'll likely need to file a motion for relief to lift the automatic stay.

I also researched that I might be able to file a motion with the court asking for "use and occupation" for the tenant to pay rents while the case is going on... but not sure if this is applicable for IL cook county and how likely will we get the rents...Can you provide me advice on how to proceed? 

Once the wet stuff is gone you can use a dehumidifier, run the ac ( expensive w/o insulation, but what are you going to do) or even box fans to dry out the framing.

You can get COMMERCIAL dehumidifiers and blowers off of ebay in package deals. 

Open it up and run fans and dehumidifiers for about 8-12 weeks and you will start to get to the moisture content levels that allow you to start replacing insulation and closing walls.  

My one struggle is that after watching plenty of videos reading the books and doing research, I feel like I don't know where to focus my new knowledge, and how to get that first deal in motion.