Why the characters 'Walter White' and 'Jesse Pinkman' dropped from the spin-off 'Better Call Saul'?

It's a little facetious to say that Walt and Jesse were dropped. They were never intended to be lead characters. The show is about Saul Goodman (hence the Better Call Saul). Of course, they could make cameos, as Vince Gilligan, the Breaking Bad creator, said:

“Personally, I’d have a hard time resisting putting all these guys in for a cameo or two every now and then.”

However, the show is focussed on Saul. The actor who portrays him, Bob Odenkirk, commented:

“Here’s what I told Vince and Peter – Pete Gould is also writing that show –we’re gonna make it fresh. It’s not gonna be Breaking Bad 2.0 or whatever. It’s a different show and it’s gonna have a different energy and we’re not gonna try to just extend Breaking Bad. That can’t be done. That show is great, it’s done, it’s all wrapped up and delivered, and now we’re gonna do something fresh and new with a character that is rich and funny, and everyone’s excited about it. And we wanna make sure the audience sees right away that it’s not a continuation of Breaking Bad, but rather its own thing that hopefully is gonna be entertaining on its own terms.”

So to summarise - the show is simply not about them. It's about Saul. They may or may not make cameos (both have expressed interest), but they are not the focus of the show.

Added to the other answers: Jesse Pinkman would still have been a school student (making facetious art about his teachers, as well as the occasional wooden box), and Walter White a mild mannered chemistry teacher (who was giving his less attentive students an F and an inferiority complex).

Why would either of them need a lawyer (at that time)?

The other answers summarize things well: This is about Saul and it happens well before the Walter and Jesse characters even become interesting.

However, regarding "Is there any plans to include them in the future?" the fact that Better Call Saul is a prequel does not mean it may not eventually catch up to the time frame that is breaking bad, or offer us some flash-forwards in time. There have been hints of this in the media.

From mid-2014:

According to actor Aaron Paul, he has had “serious talks” with Breaking Bad creator and Better Call Saul producer Vince Gilligan about a role on the series, something that seems to appeal to Paul’s sense of loyalty and a want to revisit the version of Pinkman that was unspoiled by Walter White’s “empire business.”

From February 2015:

[Bob Odenkirk:] "I'm sceptical [sic] of Vince's denials," he told The Independent in an interview. "He does know where he's going to use those guys [Walt and Jesse] or have them appear. He just doesn't want to say it yet."

The show is about Saul, many (seven) years before the events in Breaking Bad.