To answer your explicit questions:
- Hashing many passwords with the same salt is definitely a security issue. The whole point of a salt is to never hash different passwords¹ with the same salt.
- When passwords are involved, there is no way that a SHA algorithm can be more suitable than a password hashing algorithm such as Bcrypt or Argon2. Despite the similarity of names, hashing algorithms and password hashing algorithms have very different properties. Recommended background reading: How to securely hash passwords?
Now, on to your system design. It's difficult to tell exactly what you're doing, but there's probably something you need to change in your design. It may not need to be a drastic change, probably just making some changes to how you use the password and what information gets transmitted between the client and the server.
The salt needs to be unique, but it doesn't need to be secret. If the server needs to make a calculation involving the salt, either the salt should be stored in the server's database or the client should send the salt to the server.
The user's raw password is used for cryptography purposes
A password can't² be directly used for any cryptography purposes, for the same reason a cryptography hash can't be used for passwords. To turn a password into something that can be used for cryptography, such as a key, you need to apply a key stretching algorithm to the password.
Key stretching and password hashing are very similar. In fact, you can take a password hashing function with a suitably long output, split that output in two, and use one half as a key and the other half as a password hash. Knowing one of the halves of the output, you can't find the other half (unless you manage to guess the original password). You need to be very careful not to mix those two halves!
¹ By “different passwords”, I mean passwords for different accounts, or successive passwords for the same account. I'm not referring to whether the passwords happen to be identical character string.
² “Can't” meaning “your program may work but it will be grossly insecure”.