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I don't see any manufacturers recommendation for a service interval. There are ,however, intervals for inspecting the fluid. I would have your shop explain why it should be changed and if youn don't trust their recommendation then you might want to find a new shop you do trust.
Are they getting into the interior of the car or just the engine compartment? You can't keep them out of the engine area of the vehicle any other way than by somehow keeping them away from the vehicle. As far as the interior there are several ways they can get in no matter what but one way to help shut them out is to set your ventilation system on recirculate when you shut off the vehicle. When the system is in recirculate mode it closes the door to outside air which is an easy way for rodents to get in. Your owners manual should explain in the heater/air conditioner controls section how to select the proper mode.
Has the fuel pressure been tested when it will not start? With no codes pertaining to the complaint I would immediately lean towards a fuel pump issue, thats not to say that there are no other possibilities.
Have the car inspected by a reputable shop. If the car has been maintained well and the price is right, it may be a great car. The Lexus is very similar to the Toyota Highlander which has proven to be a reliable car.
Is your check engine light on? This symptom could be caused by a clogged fuel tank vent system which should trigger a check engine light and related code. If the check engine light is on or has been on, you may want to have the vehicle checked for codes to see if there are any for the evaporative system.
It depends on if the squeak happens when the suspension moves up and down or as the tire/wheel rotate. The first would be a suspension noise and the second would be a brake, wheel bearing, or seal noise.
Replacing the headlight assembly is usually the best way to repair a moisture issue.
With the key in run your fuel injector connector should have power on one wire and a ground supplied by the ECM (Engine Control Module) on the other wire while cranking. You need to determine if there is power present, if not then you need to check the power source for the injectors. If you are not getting the ground from the ECM while cranking then you could possibly have an issue with the crankshaft sensor that the PCM uses to "trigger" the injectors. You could also have a wiring issue, ECM issue, security system issue, or another sensor causing a problem.
Keep it simple. Forget the computer controls, fuel pumps, injectors for the moment and start with the basics. All an engine needs is fuel, fire(spark) and compression to run. They also need to come together at the right time. You need to verify you are getting fuel to the spark plug (emphasizing TO THE PLUG (is the plug wet or have you tried spraying something like carburetor cleaner into the throttle body while cranking the engine?)). You need to verify that the spark plug is firing(emphasizing THE ACTUAL SPARK PLUG needs to fire in the cylinder). You need to verify that you have enough compression to ignite the fuel (a compression test). You need to verify that the spark plug is trying to ignite the fuel at the right time during the compression stroke (a timing test, a tad more difficult but needs to be done). If all the above is occuring correctly the engine will start. One of the above is not occuring correctly and that gives you a path to troubleshoot.
Sounds like you need a battery/electrical system test. A $20 test can usually pinpoint the problem and maybe its something simple like a loose or poor connection.