Three questions:
1) What was Barzini's strategy when Sollozzo went (or was sent) to the Corleones? If Sonny hadn't made his mistake, there must have been another idea in Barzini's mind. And what if Vito had agreed?
2) At what time did Barzini talk to Bruno Tattaglia? The novel tells us that after Sollozzo's visit Luca went regularly to Bruno's bars. Was Bruno instructed by Barzini in advance? Or did he call Barzini: "Hey, that old Luca is hanging around all the time in my bars, what shall I do with him?"
3) If Vito hadn't been slippin' and not sent Luca, what would Sol have done to stop that killing machine?

1. Killing Luca was part of a master plan. They would kill the Corleone's best hitter and the Don in one stroke, thus crippling the family. Vito made a mistake sending Luca in the first place. It was well known he was extremely loyal to Vito, and he was hardly the best person to bluff his way past Tatt Jr. and Sol. I have always thought at the moment when Luca refused to shake hands on the deal he made with Sol, it was a signal that he was not being truthful. Sonny's "mistake" demonstrated a division in the family over the drug trade and it solidified the idea that if Vito said "no," Sonny might make the deal with him out of the way. Had Vito accepted, I believe they would have honored the deal, at least until Vito died, ad which point they would start chiseling into the Corleone family territories.

I don't understand the third question.

What DT said on the first 2 questions.

I think the third question is how would Sollozzo had dealt with Luca, had the Don not tipped his hand and sent Luca into the lion's den so to speak.

So the Don is shot and we have an enraged Luca going on his roaring rampage of revenge...

I think that one way or the other just as Sollozzo and/or Barzini had already penetrated the Don's security, Sollozzo would have already devised a plan for dealing with Luca. The fact that Luca came to see him just made things easier for him.

But they would have removed Luca from the board by hook or by crook before they made their move on Vito.

i think that vito knew that a move on his life was going to be made and having a monster like luca around when it does would not help the situation. even after him being shot and the murder of his son he still wanted peace. but would luca allow this? i do not think so. maybe and this is a big maybe, vito being the great don who wasnt dumb, maybe he knew this would happen to luca... maybe he sent him to his death.. knowing that if they were succesful in killin him that luca would make more problems for the family and further hurt the corleones...

like i said big maybe

This was a case of the Don not telling his consigliere everything. Tom had no clue that Luca had been sent as a ouble agent, and he tells Sol that he may calm Sonny, Tessio and Clemenza down, but that there would be nothing he could do to stop Luca Brasi. In a great bit of acting, Sol smirks and says something to the effect, well let me worry about Luca. Of course he already knew Luca was swimming withthe fishes.

BTW Tom's tipping his hand by mentiioning Luca's fierce loyalty gave Sol and Barz another insight into the Corleone operation's lack of communications.

Perhaps Vito's sending Luca was a tipoff as someone has pointed out. Just how much did that action influence Barzini's and Sollozzo's decision to murder Vito? That, in turn, calls into question how much a role Sonny's outburst influenced them.

Sorry for my English today. What I mean is: How would they have killed Luca, if Vito hadn't sent him to Bruno?

I've always believed that Sol went to Barzini first. Barzini knew that Vito would say no if he knew that Barz was Sol's partner, so he advised him to go to Tattaglia: "Tattaglia's a pimp--Vito's not afraid of him." Barzini had nothing to lose: if Vito said yes, Vito's capital and police/political protection would help make Barz, Sol's hidden partner, rich and powerful. If Vito said no, Barz knew that Sol, as a self-respecting Sicilian, would try to kill Vito. If it succeeded, Barz would be on top. If not, the Corleones would take their revenge on the Tatts. So, Sonny's gaffe or no, Barz thought he had his bases covered.
2. Because I believe in the above strategy, I think Barz talked to Philip Tattaglia just after Sol approached him.
3. I believe that Vito sending Luca to make nice to Bruno was the ultimate signal that the plan should go forward. It was so transparent that Barz, Tatt and Sol knew Vito was running scared--and not thinking.

Right.
Luca was Vito's Psycho for Hire and his Dragon. In a lot of books/movies and occasionally IRL, the most dangerous retainer is removed before the attempt on the big boss. If Luca hadn't come calling, Sollozzo would have had people take down Luca at his home. In the book, Luca realizes from a comment Bruno makes that the Tattaglias and Sollozzo have been watching him and verifying his schedule.