Baby budgies beg their parents or clutch siblings for food when they are not yet fully weaned. Make sure your little one is filling his crop and eating well on his own, as he grows more, the begging will stop.
It is a good sign that your older budgie preens and sometimes feeds Kiwi, it only shows they are bonding well and like each other.
The occasional pull of the tail and the playful light bickerings are also normal between budgies, as long as they don't play rough and hurt each other, then all should be fine.
Keep on allowing them to play and hang out together and let them set the pace of their friendship, instead of imposing it on them. Most likely they will choose to move in together to the same cage on their own accord, when one goes inside the cage and the other follows.