That breaker needs to be 15A, for starters
Since there's 14AWG wire in the circuits, you'll have to replace that 30A breaker with a 15A breaker to give this situation any hope of being safe. This would be the first thing I'd do, even, before fixing any other issues with the circuit. The good news is that for most panels, a 15A breaker is readily available at the home store, although you will need to match up the labels on the breaker to the labeling on the inside front of your breaker box. If you are having trouble matching the labeling up, the labeling on your box is missing, or your breaker box has breakers that have single tabs on the back instead of pairs of "jaws", post back and we can help you here.
Working our way along the circuit...
The easiest way to approach this once you have the breaker swapped is to work your way along the circuit from start to finish, fixing issues as you go. We start with where the cable exits the electrical panel, which needs a cable clamp (one of the plastic "button" style clamps will do just fine) to prevent the cable jacket from getting torn up by the edges of the hole. Moving on, you'll then need to box up the open splices, as you indicate; single gang plastic old work boxes installed into the garage ceiling will work just fine for this job, although you'll want to redo the splices with fresh wire nuts of the appropriate size. Make sure to write down which wires are connected to which before undoing the splices, though!
Finally, we get to the red and white wires standing alone, which violate several sections of the NEC. You'll want to rip them out and replace the red wire with a proper NM-B cable, as wires by themselves need to be in a conduit or other sort of raceway to be used legally under the NEC. At the switch box, the black wire in the new cable goes to the screw on the switch that the red wire went to, while the white wire in the new cable goes to the spare neutral screw on the receptacle, and the bare wire in the new cable gets connected to any and all existing bare wires in the box. Don't forget to make sure the cable is clamped when running out of the box!
As you run the cable, you'll also need to make sure it's at least 1.25" back from the edge of the stud so it doesn't get nailed into by accident, and secured with a staple or clip every 4.5' as well as within 12" of the boxes at each end. You'll also need to enlarge the hole in the top plate the old wire ran thru so that it will fit the new cable, of course.
Finally, when you reach the light box and run the cable into it (use a new cable clamp if needed), you can wire the black wire to the hot terminal or wire on the fixture, the white wire to the neutral terminal or wire on the fixture, and the bare wire to any green or bare wires from the fixture if present, as well as to a pigtail to a grounding screw into the back of the box if the box is made from metal. Last but not least, the holes that the old white wire went through will need to be plugged with knockout plugs, replacing any cable clamps that once held the white wire in.