Verified answer

First question: Is ∑ (a_n + b_n) = ∑ a_n + ∑ b_n if ∑ a_n and ∑ b_n are divergent?

Answer: Negative -- Here is a counter-example:

∑ ((-1)^(n+1) + 1^n) = ∑ 0 is clearly convergent to zero, but ∑ (-1)^(n+1) and ∑ 1^n are both divergent because the general term for either series does not go to zero as n goes to infinity.

Second question: Do a convergent series and a divergent series always sum up to a divergent one?

Answer: Affirmative -- Here is a quick indirect proof:

If ∑ a_n + ∑ b_n = ∑ c_n where without loss of generality ∑ a_n is divergent and ∑ b_n is convergent, then write ∑ a_n = ∑ c_n - ∑ b_n (*). Now if ∑ c_n is convergent, the right hand side of (*) will be the difference or two convergent series, which in turn will be convergent. However the left hand side of (*) is divergent by assumption. The contradiction shows that ∑ c_n must be divergent.