Yes, you can bring in a soft sided cooler.
What to hit first is a big question mark, because of the ages of the kids. The 8 and 12 year old may make the height requirements of the big coasters. The little kids will want to stay in the play grounds for a long time. You may want to split up and set meeting places. The husband taking the older kids on the coasters, and you baby sit the smaller kids in the play grounds.
Visiting near the end of September is good, crowds are light, so if you go in the evening, things should be ok. Only thing is, the park closes early at 6 or 7pm, so your plan would be to arrive early at 8am, and then no one is there.
Haven't stayed at the Days Inn maingate, and most of the time, it's luck of the draw when it comes to getting a good room.
The trolley doesn't stop at Universal, but across Kirkman road in the TGIFridays parking lot, so I don't know if that will work for you. Renting a car is a good idea even if you have to pay for gas and parking fees. The convience far outweighs the costs.