How can I level my basement floor, which has a 4-inch dip?
You may want to google "concrete leveling" in your area.  There are companies that will come out and pump concrete under the floor to raise the level back to where it was.  This will likely be a better solution, and may prevent further settling in the future.

For your "bonus" question:  A cheap bottle jack with a 4x4 should be sufficient to lift the studs back into place.  

A side note:
You may want to think about getting a structural engineer out to your place before you attempt to fix either of these problems.  The large amount of settling may be pointing towards a larger problem that you would be masking by just leveling it off.  The beam sagging is also alarming.  If there is too much load placed on that section of the floor, you may have other (and perhaps larger) structural issues that may or may not be readily apparent to you.  Also, the floor which will support the new beam may not be sufficient to handle the extra load from above. 

It is likely worth the couple of hundred dollars to make sure your repairs will, in fact, fix the problem instead of just masking it or creating larger problems.