There will be a 4th movie. Those who already are mentioned to be involved are very eager to see the movie take flight. Their status of involvement has also been changed from Rumored to Announced.

Time to reboot again and remember what made Star Trek the cultural giant that it was. This cast has run it's course.

125mil is a good budget for any film not too high and not too low, it's ridiculous how much money studios spend these days.
"Some of the worst things imaginable have been done with the best intentions"

i don't know, keep the budget 125m, and they can do a sequel

I am glad they are still pushing forward with the fourth movie, I was initially worried that Paramount would shelf it following the failure of Beyond, but it seems that the studio is still very keen on making more movies for Star Trek.
But I hope they will lower the budget for the next movie, Paramount had spent too much money on Into Darkness and Beyond, spending less money on the production would increase the potential profit the next movie would make.

Simon pegg and doug young are currently working on the script
"Some of the worst things imaginable have been done with the best intentions"

Yes, Chris Hemsworth will be returning

It really depends since it flopped. But like the other guy said if they do its going to be a much lower budget from this point on and that could mean some of the cast not signing up again because I'm sure they are going to do it for what they were paid on this film or even lower and only two of them are under contract for the fourth one. So I won't hold my breath unless they can get all the cast members to sign up again and yeah much much cheaper.

Remains to be seen. If they do it will surely be with a reduction in the budget.