If there is forage, there is no need to feed if you are happy with their pace and performance. However, if your objective is population build up, drawing of a lot of comb, and rapid expansion of the hive. Feed is essential and will make the difference between a populous hive hive filling honey supers in 6 weeks versus not feeding and a hive that is still working on drawing out comb in two months. If you want them to build fast, feed so they are never slowed due to lack of resources and gets them over the hump until there is a strong foraging force. If you are happy with their pace, do not feed.
Beekeeping is regional in a sense of what season you are in, the forage available, climate and weather. Beekeeping is universal in a sense of what is going on inside the hive and what the bees need at each stage of their annual cycle to achieve their maximum potential.
I think there is more than enough valuable information throughout above posts to disseminate what information you need to take away to accomplish what your goals are with respect to getting a new hive established and drawing out comb.
On your questions;
- no, brood rearing takes priority. They will not crowd out the queen from the feed - provided you are attentive and are adding space appropriately
- no, you will not have syrup feed in your honey supers because logic and reason says that if honey supers are going on then they do not need feed anymore and the feed has been removed prior to adding the supers.
Have fun with it!