Followed by vehicle, who was acting suspicous. Turned right on highway access road and the vehicle following me turned right across 3 lanes of traffic to follow me. Vehicle behind me hit their lights and with fake police today, like there are, I raced home (which was 3 blocks away.) I figured if it was not the police, I could get home before them and they would not know where I lived. If it truly was the police, they knew where I lived and would come to my house. I pulled into my driveway and waited in the car. The police did show up and approached me asking why I ran from them. I told them just what I said above. The sheriff then said, well you were following all traffic rules until you took off. The sheriffs then ordered me out of the car, asked if there were any drugs in the car, and then proceeded to search the car. When my roommate heard the sheriff yelling out front and saw the lights, she opened the front door and the sheriff told her to not come out of the house and to shut the door. I was arrested for "evading arrest" and after the police car that I had been put in left my house, the sheriffs had my car towed. CAN THEY DO THAT?
Assuming, pending information about the jurisdiction, that this happened in the US, the answer is it depends. As described in Justia's Vehicular Searches article and Wikipedia's Motor vehicle exception article, Under US law police can search a motor vehicle without a warrant if they have probable cause. This doctrine was first stated by the US Supreme Court in Carroll v. United States 267 U.S. 132 (1925) and later modified or repeated in Husty v. United States, 282 U.S. 694 (1931); Scher v. United States, 305 U.S. 251 (1938); Brinegar v. United States, 338 U.S. 160 (1949). All of these cases involved contraband, but in Chambers v. Maroney, 399 U.S. 42 (1970), the Court broadened the rule to evidentiary searches. See also Coolidge v. New Hampshire, 403 U.S. 443, 458–64 (1971)
In Preston v. United States, 376 U.S. 364 (1964) the Court ruled that the search must be reasonably contemporaneous with the stop, so that it was not permissible to remove the vehicle to the station house for a warrant-less search at the convenience of the police.
Subsequent cases have widened the scope of the exception, although the probable cause requirement has remained, and stops must have at least "articulable and reasonable suspicion". In the case of an ordinary traffic stop, the police may conduct a Terry-type search, based on "articulable suspicion". However, if the police arrest the driver, that is grounds to search the car under Atwater v. City of Lago Vista, 532 U.S. 318 (2001) and New York v. Belton, 453 U.S. 454 (1981)
Under Michigan v. Thomas, 458 U.S.(and Chambers v. Maroney, 399 U.S. 42 (1970); Texas v. White, 423 U.S. 67 (1975); United States v. Ross, 456 U.S. 798, 807 note 9 (1982).) police may remove the car for a search elsewhere if they have probable cause to believe that there is contraband (such as alcohol, drugs, or weapons) in the car.
Thus the arrest justified a search, as described in the question. It might not have justified the tow if there was not probable cause to suspect contraband. But that depends on the detailed facts, and judging those is beyond the scope of this site.
In Collins vs Virginia (2018) the Court ruled that a warrant-less search of a car parked on the owner's property and within the curtilage was not justified. (The "curtilage" is the area near the house which gets additional Fourth Amendment protection, almost as if it were inside the house). However, in that case no arrest was made at the time of the search, and the search was based on an officer recalling the car having been used to elude arrest some two months previously, not just a few minutes before, so that case may not apply in the situation described above.
If there was no probable cause, any evidence found might be subject to suppression, and a suit for damages under sec 1983 might be brought against the officers and the police force, but that would depend on the specific facts, and would in practice require a lawyer skilled in such matters.