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Depressurize fuel system by switching ignition off, and then loosening the fuel cap. Remove the rear seat cushion and the service cover. Disconnect the fuel pump harness connector. Start the Eclipse and allow it to run until it stops. Turn the key off and crank the engine for 2 seconds or so. If the engine does not start, turn the ignition off. If the engine starts, continue starting it until it does not start anymore, and then turn the ignition switch off. Disconnect the negative battery cable and remove the battery and the air intake hose. Disconnect the fuel lines from the fuel filter. Remove the clamp and hose from the fuel pressure regulator. Take off the fuel filter mounting bracket bolts and then the fuel filter. Unscrew the bracket screw and remove the fuel filter from the mounting bracket. Place the new filter in the mounting bracket and attach it with the screw. Reinstall the fuel filter with the bracket mounting bolts. Reconnect the main fuel pipe to the fuel filter. Tighten the flare nut. Reconnect the high-pressure fuel hose to the filter and tighten the bolt. Install the battery and the air intake hose. Connect the negative battery cable. Start the engine of your Mitsubishi Eclipse and check for leaks.

Check all relays and fuses, some relay getting weak do this drama every morning. If not check your spark plugs, cables and coils for proper current and spark supply. Later you can check air filter and exhaust system in detail

Yes you can install any stereo if size permits, better get it done professionally as it has risk of doing wrong wiring and causing heating the cable and burn them.

Either steering stabilizer is faulty or steering centering alignment is out or loose. If in warranty let Habtoor do it free as its 2009 model, if not get to some decent wheel alignment place and they can diagnose it properly and advise you.