There is no way to add without carry. 0 is shifted into bit 0 and the original bit 7 is shifted into the Carry. The S and V flags are set to match bits 7 and 6 respectively in the value stored at the tested address. See the notes on the Program Counter for more on displacements.
Branches are dependant on the status of the flag bits when the op code is encountered.
A branch not taken requires two machine cycles. Add one if the branch is taken and add one more if the branch crosses a page boundary. One of the best flags to use for this purpose is the oVerflow which is unchanged by all but addition and subtraction operations. If the value in the accumulator is equal or greater than the compared value, the Carry will be set. It does not signal the presence or absence of an interrupt condition.