How to correctly manipulate the I/O pins of PIC chips?
PICs have the annoying gotcha of having ANSEL default to ones, which means that ports that can act as analog inputs are so by default.  This will force digital reads to zero (see port diagram below taken from the PIC16F833 datasheet to understand why), and that is probably why RB0 gets cleared when setting RB1 with a Read-Modify-Write operation.



In main, clear ANSEL before accessing PORTB and try again.

By the way, LEDs usually drop more than 0.7V, ~2V is common, so it is possible that the first experiment succeeded for the same reason that the second one failed.