Re: queen cells
A lot of the way things are done in beekeeping is management choice. Some would purchase another queen while others would let them raise their own. Those decisions are made from weighing various factors. Along that vein, I ask, why do you have a beehive? Do you want more beehives? Do you want honey? do you plan on getting enough hives to provide at least some income? These factors and more might determine whether you let them raise one of their own or you purchase one for them. Keep in mind, they are rejecting the purchased queen that they came with, they could also reject a new given queen.
Also, as an example, I purchased 2 packages last month. One package is doing fine, very solid sealed brood pattern and bees are working happily. The other, the queen was questionable from the start. She took too long to start laying, the bees were picking on her. I am not surprised that today I see superscedure cells built.
I am of the opinion that bees are more accepting of queens they raise themselves, so long as the bees are healthy and there is incoming stores to feed and raise a new queen in good condition. The hive I have, I'm letting them replace the queen themselves. I feel they will do better raising their own than they would be by me purchasing another to give them. By the time a purchased queen arrived and I was ready to introduce her, they will have already had a queen emerge. Some new queens can take up to 10 days to start laying, so purchasing a queen and giving them is not going to same me any time of no brood rearing, or not save me much. So for all the considerations I try to keep in mind, I am letting them raise a queen of their own.
Sorry for all the rambling. I will not say you should or should not do it one way or the other. Do your own research and see what others say about it here. Then make your own decision and make a choice of which way to go. Either way you do it, Best of luck and enjoyment of beekeeping.
"A good day is when no one shows up and you don't have to go anywhere." - Burt Shavitz (Burt's Buzz)