Conceptually, why would you add the cash paid for the target firm to value the cost of synergies?
If the value of the combined firm is less than the pre merger value of the acuired firm and acquiring firm, how is there any synergy?
Where did you see adding the cost? I haven’t encountered it so far. Do you have an example to work it out?
Post merger value for the combined firm is = Value of the acquirer + Value of the target + Synergies - Cost (if paid in cash)
The target will gain the difference between the Cost and its value (called the premium)
The acquirer will gain the difference between the synergy and the aforementioned premium
About the second point, that’s straightforward I think. If the cost was greater than the realized synergy you will just end up with an after merger that costs less than the sum of parts.