Assuming that sum of probabilities for all possible events that can occur should sum to 1, how does one denote this for a conditional probability? Is it $P(A|E_1) + P(A|E_2) + ... = 1$, where $E_i$ is a specific event to be conditioned on? Or is the answer something else entirely?
If $E_1, E_2, \ldots$ is a collection of disjoint events whose union equals the entire sample space (exhaustive), then $P(A\cap E_1)+P(A\cap E_2)+\ldots = P(A|E_1)P(E_1)+P(A|E_2)P(E_2)+\ldots = P(A)$.
This is a good example where intuition on conditional probability can be a good check of the formula. Suppose $P(A)=0$ (which is clearly possible); then we know intuitively that $P(A|E_i)=0$ for any $i$ (because event $A$ never happens, so how can it happen conditioned on something?), so the sum you gave will be $0$, not $1$.
But if the $E_i$ are a partition of the sample space, we do have the formula $$ P(E_1|A) + P(E_2|A) + \dots + P(E_n|A) = 1$$ with an intuitive explanation. Given that $A$ occurred, it's still true that one of the $E_i$ must occur, so the total probability of the $E_i$ occurring given $A$ must still be $1$.