solve problem: dy/dx = e^(x+y)... (make final answer e.g. y= something)

After integrating 1/e^y with respect to dy and e^x with respect to dx, and without including the constant yet, i got

-e^(-y) = e^x

where -e^(-y) is from integration of 1/e^y with respect to dy.

If i write -e^(-y) + C = e^x, I will get final answer as y = -ln (k - e^x), where k = -C, which is the answer from the textbook.

Also if I write -e^(-y) = e^x + C, I will get the same correct answer.

1. My question is that.. is the C (constant) the combination of the constants of the integrals of LHS and RHS? And therefore we only add C to either side of the equation and not both?

2. No matter which side I place the C at for any differential equation problem, will I get the same result?

And lastly.. must I include a modulus sign after a log or ln? e.g. ln | 2+x | ?

or just ln (2+x) will do? Although 2+x must be a positive value..?

Please clear my doubts.. MILLION THANKS!