If the Prince comes from CSM, it only interacts with CSM rules. You don't even glance at the Daemon Codex for a CSM model. But, you do get Daemonology from the core rules and the new powers from Traitor Legions. Traitor Legions also has new powers for the gods, so if you have that book available, you should use those for the dedicated god powers.
I'm not intimately familiar with the Daemon Codex so I may have missed something somewhere in it, but I don't see anything that lets you take a free Primaris. You usually don't get them unless you roll all of your powers on that table. Daemons can never do that since they 'may roll up to half of their powers on the corresponding chart' for the God powers. I suppose that if you have ML1, half of one is one after rounding up so yes, you can take your roll and Primaris no problem, but other than that example, they would roll their one power and can choose to replace it with the primaries via core rules for generating powers. Because Daemons is an old Codex though, they do have access to summoning Daemonology out of the core rulebook.
I don't have a big list of things but a common trick is charging two units in. Charge first with a worthless unit, or even a single model unit. The enemy gets one single Overwatch, and no Overwatch if they are in active combat. So either they shoot and maybe kill the one guy, leaving your actual important unit free to charge without worries, or they might be engaged with the one guy and your other unit can again charge in without worries.