Lot's of beeks re-queen about now. I'm doing one myself. The pros are that you should end up with a young, producing queen going into the winter. If the hive is healthy when you re-queen, then she is ready to rock and roll in the spring. You have less to worry about regarding an aging or spotty queen that fails exactly when you want her to be giving you lots and lots of spring bees. Some beeks feel that a young queen may help slow down swarming although I've never experienced that. Re-queening in the fall may also give you a break in the brood cycle earlier than normal in the fall if the new queen takes a while to get going. That can give you a drop in mite counts earlier than you would typically get. Finally, if your old queen is slowing down, but the temps are still good for laying, then the new queen might pump up fall production of brood giving you more bees to cluster for the winter. You can see that a lot depends on your colony. Of course, if you're just switching from one type of bee to another, fall re-queening lets you go into the winter with, say, Italians and start the spring with, perhaps, Carniolans.
There are some cons. What if your old queen is just....old. Replacing her with a store bought queen might give you a poor queen at a time of the year when you may not get another chance to re-queen. The girls might not accept the new queen. You might cause damage when you handle her. Worse, you may not know if she's a good one until the spring, when things should be booming. These are all risks, but managable ones.
The best method for re-queening is kind of up to you. Nothing in your post is wrong. If you can find your old queen and pinch her...leave the hive queenless for a bit....and then introduce your caged queen, you can easily check to see how they're reacting to her in a few days. I would lean towards push in but again, be careful about the handling issue. In fact, you may just want to bank your old queen as a precaution until your new queen has been accepted. As far as the honey flow goes, you can feed if you want. Some beeks feel that feeding when introducing a new queen makes the bees like the queen that seemed to have brought them food all of a sudden. I don't know about that but feeding probably won't hurt.
One last thought. If you have a weak queen that has caused a weak hive along with a strong queen in a strong hive, you may want to consider pinching your bad queen and combining for the winter. Come early spring, get yourself a store bought queen and do a split. That might help protect all your bees, cut down the spring swarming tendency, still give you a nice, new queen for the spring and probably not really affect your ability to gather honey.
Just my thoughts.....
"My wife always wanted girls. Just not thousands and thousands of them......"