Did they ever do research, investigation, independent recount to find out if Gore truly won the state of Florida in the 2000 election?

Yes, they researched, investigated, and did independent recounts.   It was determined that Al Gore would have been the winner of the 2000 presidential election, if The US Supreme Court would not have butted in.  

Yes.  It turned out, to everyone's surprise:

1. If the courts had allowed what Gore had request: a recount in those counties where he want one, and not in those counties where he didn't want one, then Gore would still have lost and Bush would still have one.

2. Ironically, if they had recounted the entire state, which neither candidate asked them to do, but the Florida Supreme Court thought should happen anyway, then Gore would have won, and Bush would have lost.

So Gore technically should have won, but if the courts had done what he asked, then he would have lost.

You're under 40 years old is my guess. There were books and movies about the 2000 election. It's not like 2020 at all. It's too late to matter whether Gore would have won. The ballot method and State Government were tainted and the results were within 2000 votes that came down to a hundred or two in millions. 

The ballots were tainted by handling in the first recount, and got worse from then on. The format of the ballots was also messed up. SCOTUS in later years was wrong to use their 2000 decision for anything afterwards. It was supposed to be a one-time deal to stop a forever recounting of things that could never be fixed.

You are using opinions rather than trying to see it neutrally.

Gore got cheated, but not by the voters or the court. He got cheated by the ballot itself and State Government in Florida.

I knew people who lived there. the ballots were defective in format and type. Look up "butterfly ballots" and "hanging chads". That's what cheated Al Gore.

(And, Governor and Secretary of State of Florida in 2000)

They did that right after Gore claimed the ballots were too confusing for his supporters, and was laughed out of the Supreme Court.

No, the dems meekly accepted their fate and Al Gore started a global warming panic business.

"They" did do the research.

Back in 2001, a consortium of media organizations including the New York Times, The Wall Street Journal, The Tribune Company, The Washington Post, The Associated Press, The St. Petersburg Times, The Palm Beach Post and CNN. went back and recounted all the votes. 

The verdict was Bush still wins even if the US Supreme Court had not stepped in and the votes continued to be recounted as requested by Gore and ordered by the Florida Supreme Court.

See the link.

They did the best they could and every count was in favor of Bush.