1998 Jetta P1582 and P0300/P0301/P0302 Misfire Codes - Stuttering on Hills, Starts After Delay, Exhaust Leak Concern

My 1998 Jetta has been experiencing starting issues after rain. It starts fine initially but begins to stutter and putt-putt when driving, especially uphill. If I floor the accelerator without stopping, it coughs and eventually restarts, with the check engine light briefly illuminating (typically showing P0302 or P0-300 misfire codes). When the car dies, it cranks but won't start for about 15 minutes before restarting—after which the check engine light comes on and then turns off again. A few days ago, the issue persisted with four diagnostic codes: P1582, P0300, P0301, and P0302. I also noticed a noticeable exhaust leak sound when braking with the clutch, and upon inspection, found a catalytic converter leak. While I plan to replace the cat converter, I'm unsure whether this is directly related to the misfire codes or if it's contributing to the problem. Any guidance on potential causes or troubleshooting steps would be greatly appreciated—especially for diagnosing the root of these persistent P0300 and P1582 errors in a 1998 Jetta.