To get the Materia from the rocket, the lock's combination is circle, square,X,X
To open the safe in Shinra Mansion, you need to press, Right 36, Left 10, Right 59, Right 97. If your time runs out, or if you go past the numbers, you need to start over. If you do this correctly, the boss will come out.
How To Get Yuffie and Vincent:
Yuffie
To get Yuffie, travel around forest for a long time.
Enventually you'll encounter her just like a random creature fight. After
defeating her, you'll go to a screen that has a save point. The key to
success is that you don't take your eyes off of Yuffie. Don't use the save
point, if you use the Save point, it is just a fake,
Yuffie will take some Gil( about 200 Gil), she will leave, and it does
not even save. Examine her and go through the dialogue. She'll challenge
you to a fight.
Respond not interested.
Yuffie: (She'll taunt you again).
Cloud: " petrified."
Yuffie:( starts to leave but stops).
Cloud:" wait a second."
Yuffie:(seems to come around).
Cloud:"that's right"
Yuffie:( joins the party).
Cloud:" let's hurry on."
Vincent
Vincent is locked away in the Shinra Mansion basement.
The door to his chamber is locked, but the key is easy to find. Unlock
the safe on the top floor and get the key. Open the door and go to the
casket which he sleeps in. Talk to him several times. When you leave the
mansion, he'll agree to join your party. Simple, but first you need to
open the safe.(see top for combination, you have to beat theboss before
getting Vincent).