Multiple issues

I purchased this car about a month and a half almost two months ago the car has had a total of three owners I am the third the second was my brother the car has 152 something thousand miles and my brother did sell it to me cheap because I needed a car to get me to and from work and it does not have reverse. When you try to put the car in Reverse you can hear it try to engage you can hear it click and you can feel it try to engage but when you try to put your foot on the gas it does not move however if you are on a slight incline and even in my driveway that is completely flat it will roll back on its own when it is in neutral it will not roll back at all in my driveway like it does in reverse. When my brother still had the car he had someone look at it the guy he had look at it does not know anything I have read multiple reviews about him and he has cheated multiple people out of his their money that he has ruined their Motors there Transmissions rip them off he is just a bad guy all the way around he told my brother that there was metal shavings in the transmission fluid and that the bands in the transmission were bad and that even if the fluid and the filter were changed in the transmission that it would completely ruin the transmission and it would no longer run at all currently up until Saturday the car ran perfectly fine in Drive as of Saturday it started acting weird it started having issues with accelerating and then as of Sunday it overheated on me and on Monday I replace the thermostat after replacing the thermostat it overheated again we bled the line and got all the air out and also ran a toggle switch because the radiator fan was not kicking on the water pump on the car is new I know the car is not getting hot but the temperature gauge is going crazy it goes from running at regular temperature too cold as if the car was just started even when it's been running for thirty minutes or longer it has a really hard time accelerating it makes a really loud whining sound that started as of Saturday we realize that there is a hole and a crack in the reservoir for the power steering pump reservoir I do intend on getting a new reservoir on Friday I'm just not understanding what could be causing these new issues along with the issues with the reverse as well when I check the transmission fluid it does smell burnt and if it is a brown color instead of the normal red color. As for the accelerating issues when that did happen on Sunday it was as if the car would overheat as the RPMs would get higher the higher the RPM the hotter the car would get I did get it to the side of the road and pulled over before it got too hot so I know for sure I did not blow a head gasket this is the six Cavalier I have owned so I know them like the back of my hand but this one has me dumbfounded! I personally have had it looked at by four different mechanics I've had one person tell me that it's the transmission is completely bad I've had one person tell me that it just needs new fluid and a new filter I've had one tell me it's the solenoid and I've had one tell me that it is the pump that is inside the transmission. I have reached out to multiple transmission shops in my area and none of them will look at it because they state because it is such an old car and it does not have reverse that I am better off just selling the car to a junkyard and going and getting a newer car but I do not have the money to do that nor the credit. And I personally like my car and you not want to go in debt for someone else's headache I would rather fix what I have even if it comes down to having to replace the transmission I just do not want to get ripped off!