The nursery manager or assistant manager must be on the premises at all times whilst care is being provided.
Good nurseries operate a key worker system, if they have this system in place then you should certainly be aware of who it is. Your childs key worker will not necessarily be present with your child at all times, however they should be the person who knows your child best and who is mainly responsible for planning and observations EYFS etc for your child.
It isn't unusual to get feedback from several people, each nursery worker will have several children that they are key worker for, and at collection times issues may have arisen with other key workers children that require more urgent attention IYSWIM. However your key worker should be working in partnership with you as a parent and sharing DS's portfolio with you, if your child only attends 1 day per week this portfolio may not necessarily be very big (don't know how long he has been attending).
The not having coat on wouldn't worry me too much unless it was a regular occurance, same with socks, children this age quite often play hide their socks (drives me potty).
Generally their is a senior worker in charge of a room or Room Leader, room leader's must have a L3 qualification, and at least half of the other staff must also be L3, students must not be counted in the ratios and a Level 3 must be in the room at all times as should a person who is paediatric first aid trained.
The TV thing I don't particually like, although their are some good things that children can watch which help them learn and remember things. Also a lot of nurseries use the rooms for training as well and we are inundated with useful DVD's to watch so this could also be a reason. If use appropriatley it should be ok, if they are stuck in front of it every day then I wouldn't like it personally but then that's my oppinion.