2001 Mitsubishi Mirage Turns over but won't start

My vehicle info:

2001 Mitsubishi mirage DE Coupe
1.5L Engine
Manual Transmission

The Problem:

Lately I've been having starting issues. They are intermittent. The car turns over but will not start. I have to try over and over and over again before it finally starts. Once it starts, the vehicle is fine. It runs smooth and it does not cut off, nor are there any signs that anything is wrong. Once its off I can start it back up, most of the time. Today it started in the morning but it sounded like it wanted to cut off, I hit the gas and it started to idle smoothly and I drove it to work no problems. When I got off it started but immediately cut off, I can't even hit the gas, it just shuts off.

What I've done:

Recently I've been working on it and I've installed a new: battery (2mos ago), alternator(2mos ago), sparkplugs, wireset, distributor cap. The problem started before I replaced the sparkplugs, wireset and dist cap(1wk ago). Today I borrowed a scan tool and got back 2 codes P403, and P421. I looked the codes up: P403 Exhaust Gas Recirculation Circuit Malfunction and P421 Warm up catalyst efficiency below threshold. Tomorrow I'm going to test the EGR Solenoid and the O2 sensors, could this be causing my problems?

I want to try and fix it myself before I take it to the shop. What else could be causing this?