So I hope you now understand the importance of getting fault codes before attempting to replace components or taking it to a shop. You could have replaced the IACV and ECU for $1000+ and that would not have fixed the problem.
You also mentioned that the PO replaced the 'ignition system' so you now have 2 keys - 1 for the door and 1 for the ignition switch. Well... what he replaced was actually the Ignition Key Cylinder (that allows you you to turn the ignition switch). The Key Cylinder Lock can fail when users bundle too many keys together with the car keys. Is the PO a building maintenance or security officer?
As for the 'dying' remote, all you need to do is get a replacement battery -available from Target, car parts store or electronics parts store. You can buy a 2nd one if you want to but we can discuss that later. You can use a coin to remove the battery cover at back of the remote - just make sure you don't let the cover 'fly off.'
Let's look at your fault codes one at a time, starting with DTC-0440 - EVAP Control System (small leak, negative pressure. That means there is a leak somewhere between the Fuel Tank Filler Cap and the Evap Purge Line to the air intake manifold (close to the throttle body). There are other causes of these problem but the most common ones are:-loose or dirty or cracked fuel filler cap (may require proper closing of cap, cleaning or replacement of filler cap)-corrosion on the filler neck (just need cleaning/removal of rust)-or the PO used a non-OEM fuel cap, hence the leak (use OEM part as much as possible).
Below is the link to 2001 I30 FSM (Factory Service Manual). You need to go to page EC-20 (Engine Control System chapter) to familiarize yourself with the Emission Control component location. This is important so you get to see the engine components and sensors yourself. Then look into page EC-362 to get a better understanding of DTC P0440 and probable cause.
http://www.nicoclub.com/FSM/i30/2001/ec.pdf
The next problem DTC P1320 is more COMPLICATED for your level. This requires usage of multimeter (Volt-Ohm-Amp). Please avoid the cheap ones - aside from errors in reading, it can load the circuit. Testers from Fluke, HP and the likes should do the job. In addition, one has to use a wiring diagram from the FSM to get a better grasp of the testing involved. Use the same link above and start on page EC-506 and go from there. So if you don't have a tester or don't know how to use it, ask a friend who may be able to help you.
When you take a car to a shop for this problem, P1320, they may diagnose and correctly isolate the actual problem. However, some shops NOT FAMILIAR with Nissan/Infiniti cars will most likely replace the Ignition Coils and Spark Plugs immediately, hoping that they will fix the problem. If not fixed, you may be $800 down and the problem is still there. You need 6 each - coils and plugs. Discounted price from IoS - infinitipartsUSA.com is $78/coil and $$8/plug. Use OEM coils and plugs as much as possible. Avoid Bosch spark plugs.
Since your financial situation is tight I strongly recommend that you ask a friend to help you with the testing. At least, you will have an idea as to what the exact problem is before taking it to the shop.
Below, I'll try to make it simple as much as possible. The wiring going to connectors are color-coded so even w/o the wiring diagram, for as long as you know how to identify a color-coded wire, you can do the testing.
DTC P01320 indicates Ignition Signal in the PRIMARY CIRCUIT is not sent to the ECM during engine starting/running. Probable cause:-harness or connector between ECM and each ignition coil-ignition coils-condenser-Crankshaft Position Sensor (REF) or circuit.
We can start with the Ignition Coil, then the Harness-Connector, Condenser and Crankshaft Position Sensor.
You car is equipped with DIS (Distributorless Ignition System). The Ignition Coil (6 ea.) has a built-in power transistor whose bias (control signal) is provided by the ECM (Engine Control Module). The coil is powered thru the ECM Relay anytime the Ignition Switch is ON. The Ignition Coil connector has 3 terminals as follows:-Pin 1 (striped wire) - is where the bias signal from the ECM (@ 0.2VDC with engine at Idle) is connected to alternately turn ON/OFF the power transistor which then enables the spark plug(s) to fire or not. An oscilloscope will provide a better display of coil control signal-Pin 2 (B-Black wire) - is the GROUND connection for the coil. Make sure the engine harness ground points (top of engine, besides intake plenum) are clean and secure. Verify this with ohmmeter (@ 0 ohm if good).-Pin 3 (R-Red wire) - Battery voltage (11-14.7VDC) is connected to this pin and the CONDENSER (top of engine, besides harness) whenever the Ignition Switch is ON. If the voltage is low, the condenser may be leaking or shorted. The condenser acts like a 'shock absorber' for the ignition coil (too complicated to explain at this time). When disconnected, the condenser should read @ 1M-ohm at room temp.
The Crankshaft Position Sensor uses Hall-Effect to monitor the TDC (Top Dead Center) position so the ECM can send the appropriate ignition signal. This sensor is mounted near the crankshaft pulley with a typical resistance of 470-570 ohms across Pin 1 (Black wire) and Pin 2 (White wire). Refer to page EC-519.
So you need a functional multimeter for the following tests. The Bank 1 coils for cylinders 1-3-5 are on top of the engine, towards the firewall. The Bank 2 coils for cylinders 2-4-6 are towards the radiator and are accessible by removing the engine cover plate.
To check the Ignition Coil, disconnect the connector from each coil. Refer to page EC-515 for harness and coil locations. Then measure resistance across Pins 2 and 3 of the Ignition Coil receptacle- if reading is 0, coil is bad; not 0, coil is OK.
Next, check if POWER is available to the Ignition Coil (Ignition must be ON). This can be done with the connector in place or disconnected. Testing must be done on the connector going to the coil (and not the receptacle of the coil). With Ignition Switch ON (engine not running), check Pin 3 (red wire) of each ignition coil for presence of battery voltage (@ 12.5VDC) w.r.t. chassis GROUND. Each coil MUST have the battery voltage or it indicates problem with the harness/ECM connection. Absence of battery voltage indicates a bad ECM relay or wiring. Low voltage indicates a leaky condenser.
If the required voltage is present in each coil, then we can now check the control signal (bias) from the ECM. Ensure all coil connectors from the harness are connected to the respective coils. Start the car and check the CONTROL signal from the ECM. Each Pin 3 (striped wire) should read an average of @ 0.2VDC (200 mV) w.r.t. chassis GROUND. If not, bad wiring or signal from ECM. The presence of this voltage also indicates the crankshaft position sensor is working. Again, an oscilloscope, if available, will give a better display of 'firing' signal from the ECM.
Additional t4esting may be required such as continuity testing between engine harness connector and wiring to the ECM. That is also in the FSM.
Good Luck and let me know how it goes