everything seems like Al Neri was the Underboss of Michael in GF3, because we don´t see anybody more than him except Michael, of course until Vincent had come

Don`t know. If there only was a book to GF3.
I always thought Al Neri was Mike`s "Security Director".
That he had a very special position in the family.
And he sometimes had to go out for a special job where Mike didn`t thrust anybody else.
It was in fact a bit hard to understand the structure of the family in GF3.

He was just Caporegime.

Evolution off the Caporegimi:

Clemenza - Tessio (Part 1, Don Vito)
=>
Clemenza - Tessio - Rocco - Neri (Part 1, Don Michael)
=>
Rocco - Neri - Pentangeli (Part 2, Don Michael)
=>
Neri - Zasa (Part 3, Don Michael)

I doubt that Neri or anyone else was Michael's underboss in GFIII because by that time it seemed that he had thrown off most of the operating aspects of running a Mafia family. Though he claimed to be "legitimate" by that time, he was still an eminence grise in the Mafia--apparently serving on the Commission and generally acting as an elder statesman. But most, perhaps all, of his day-to-day activities were in the legitimate world. As he said, Zasa held the "olive oil business," so I don't think he had a need for an underboss. Neri, IMO, was just a bodyguard and general factotem.

Yea it can be argued that at different points throughout the film, different people seem to be acting as Underboss and Consigliere, just general advisors and "right hand men" to Michael, namely Neri, Connie, Vincent, BJ Harrison, Zasa, and maybe even the Abbanando kid at points.

As previously mentioned, Michael seems to have thrown off the traditional family structure as he became more of legitimate businessman and left the more criminal aspects of his business to underlings.

If I had to give concrete answers, and also assume that Michael was still abiding by the family structure, I would guess Harrison was his Consigliere, Neri was his Underboss, and although Connie held no official position seeing as she was a woman, she was also very influential and always had Michael's ear.

Harrison was Michael's lawyer, nothing more.

Neri was as close to an underboss as you can get. I just noticed watching GFIII for the millionth time this weekend that when Vincent bites Zasa's ear, it is Neri who berates Vincent while Michael just shakes his head in disgust. Neri had to have a good deal of power to berate a blood member of the family without getting Michael upset.

And don't forget with Neri, there were no buffers.

Don Tomasso, Is Harrison Michael`s lawyer? Isn`t he just a politician under Michael`s influence? Because at the beggining of the movie he always said he needed lawyers, he says that to Vincent and Anthony.

B.J. Harrison was Michael Corleone's lawyer. He was mistaken as Consigliere of the Corleone Family by alot of people because he advised Michael Corleone on alot of topics, i.e., Michael Corleone's purchase of International Immobliare. He and Michael Corleone's relationship was completely professional. He did not advise Michael Corleone or any other member of the Corleone Family on any organized criminal activities.