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Firstly, other than the filter, no one replace a few 'PIECES' on a bad transmission. Your first question and answere must ne, do I really need a transmission. Chances are the response you get will be. "we can't tell till we take it apart,' and then of course its too late to have the trans put back together for less than 900 if not needed. So if there are n simple problems, plan on a new transmission. The next question should be, Is the car worth it to me to spend $3000 on. KBB on a 97 XLS Avalon with 205K in my area in about $1900 in average condition which also means a WORKING transmission. So spend $3K on yours to have a working $1900 car? I think you might be better off using your 3K on a new car and sell the Avalon for the few hundred and consider yourself lucky. Remember, anyone buying you car STILL has to put in a trans so they face the same financial question you have. UNLESS someone who happens to have a spare working trans for your car in their garage and have to have a 97 AVALON WITH 205k miles on it finds you, you car is going to the salvage yars. So consider ANYTHING YOU GET FOR IT AS A BONUS.
It is near impossible to tell you how much a repair would be without diagnosing the problem. Anyone telling you different should be avoided. Your problem could be as simple as a clogged fuel filter, bad gas or clogged air filter. Your "rod" noise is likely engine knock so your sensor may be failing but doing what it is supposed to do, reduce power, when bad or low fuel flow is detected. Engine replacement is just as variable depending on what you buy, new entire engine, shortblock, rebuilt, junkyard, etc.'
Since its unlikely that BOTH bulbs would fail a the same time and if everything else on the circuit (check owners manual or wiring diagram) it's not a fuse. So try Googling for location of a relay, if there is one. However I'm guessing it's likely the microswitch on your trans mission that causes the revers lights to go on when put into reverse. That's where I'd start but if you are not auto mechanically inclined, see a reputable tech or so inclined friend.
Have a simple alternator test done at your parts house or mechanic. If alternator is OK then you have something drawing power when the ignition is off. A simple battery draw test can tell if you have a draw like that. If you do, the forst question is, "what extras have been added to the car ie stereo, phone charger, navigation, etc' and start there to make sure they aren't the source of the drain A 'GOOD' but marginable battery can be drained overnight pretty easily, especially in winter, by some simple electronic add ons that are left on.
Should be easy for a tech to figure out in 30 minutes. Likely are collapsed cat converter/exhaust pipe or faulty ignition coils or ignition module. Is it up shifting, if an automatic. IF not a possibly failed modulator on trans could be problem. When you post a question, the more information you post, ie mileage, transmission type, any previous diagnosis, can help and will get you more and more accurate responses.
Although you didn't say, I assume you are plugging the OBD into the lighter for power. If not, then it isn't the fuse as your OBD port does not input or take any charge from the port itself. If you are plugging it into the lighter than it is possible you are overloading the circuit, thus blowing the fuse. However if that is the case, either you have add on accessories running on the same circuit as an unaltered lighter circuit has built in reserve capacity. Does your lighter element work properly when plugged in as that's when the circuit gets it heavy load. It draws nothing when the element is not being heated unless other add on or factory accessories have been put onto its circuit. Goggle and you should find a wiring diagram and see what else may be on the circuit from the factory. If there The lighter circuit is a common one to use for add ons as, in most cars because it can be a power always on line or a switched power line depending on the car. Plus it is usually very accessible. If your lighter element isn't working, you might have a damaged receptacle that is causing a short when the OBD is plugged into it or another device is. I would also check to make sure you have the correct fuse, blow clean the fuse box, and try to check for any corrosion in the box. Do not be tempted to put in a bigger fuse as that will cause you bigger and smokier problems. If there are other factory items on the circuit, make sure they are working properly, correct bulbs, etc.
From your postings, it seems confusing if you have already bought the car or not. If not, go to your public library and check Consumer Reports. Go online and check any of the used car prices book sellers such as BLUE BOOK and NADA.
When is it on? Which "engine light?"All the time? It also might help if you have a tech check you computer through your OBD port and tell you if any fault codes are showing.
First make sure belt tension is correct and you have the correct type belt ie a "v" or flat belt. If your car has tensioner pullys, they are questionable after 50K MILES BUT can fail anytime if defective. However they USUALLY become noisy if bad. If you truly have NO noise, it isn't a bearing failure on the PS pump but the pump could still be bad due to internal problems. Check for collapsed or failing PS lines. I think you likely have a failing PS pump if your PS lines are OK. good luck.
There is no "summer" or "winter" lube for starters. Sounds like the old flubber gasket that service station people would tell a woman their car needed. Have you inspected all the teeth on the flywheel? Obviously, if you have the correct starter ( I would get a 2nd opinion on that) the starter is NOT the problem. If it is a manual, have you had any previous clutch or pressure plate work done? You pretty likely have some clearance problem that is being caused by another part of the starting assembly IF you are truly getting the correct starter. Again, I'd try another shop or parts house to make sure you are getting the correct starter. If you are, start looking at the teeth. A possible very long longshot is that the starter is not getting enough current to function properly. A rare occurrence but check the power at the starter when activated and make sure it is the proper voltage. Insufficient power due to a faulty wire or low battery may not be enough to fully retract the bendix. This could be why its happening only on cold days when the starting load on a battery is highest due to cold oil and if the battery has set overnight. Have a free 'LOAD' test done at you parts store to make sure you battery is up to snuff. Also make sure your battery has the correct CCA (cold cranking amp] rating for your car. Also check the starter wires for continuity.