I have a son in second grade this year, and a three year old starting preschool. I found last year was really hard while we tried to figure out how to handle the demands of my son's first year of all day school and homework; the hours between his arrival home and bedtime were really difficult, the kids seemed so demanding and homework was just another thing for me to juggle.
Eventually I created a "tea time" immediately after arrival home from school. I make decaf English breakfast tea, set the table with cups and saucers, and set out little dishes of snacks like slices of cheese, raisins, almonds, fruit, etc. (when we finished this "brain food" we could all have a treat, like a cookie). During tea time we would all catch up on our days, and I would ask both boys to tell me two things they did and didn't like about their days. This whole thing took about 30 minutes and basically ended up being like an entire meal. But it was sooo necessary for my oldest to have this hearty snack, and the down time to reconnect with me, and time for his brother to reconnect with him (so he wouldn't bug him so much during homework time). I gave him 30 more minutes to watch TV or play. I found he was then much more able and willing to start on his homework. Fortunatley his homework was minimal, and he usually still had time to play outside before dinner.
Overall this is still working this year, though homework responsibilities have increased,and my son is even more tired upon his return home. Thus, I have implemented a strict schedule where he needs to start his homework by a certain time. This seems to work ok, but I actually really like the idea the previous poster mentioned about letting the child choose what task to do when. I might try that and see what happens.
As for whether your child should immediately do homework or have some downtime, I think it depends upon the individual child and their habits. I personally like the idea of getting the homework done right away, but that would never work for my son - he needs a bit of downtime between school and homework. I agree with previous posters that one extra curricular activity (if even that) is enough at this age. It is just too hard to ask such a little person to meet so many commitments, and there is the risk that the extra activities will no longer be fun.
Finally, this year I started laying out the boys' clothes the night before, including shoes so that we aren't looking for anything in the morning. As much as I don't want to, I make myself get everything ready the night before, including lunches packed and backpacks ready to go. I also started this year to have the kids bathed and in pajamas BEFORE dinner (I put them in the bath and let them play while I make dinner, and then quickly wash them). This has been great, because after dinner (which is nearly bed time) there is hardly anything more they must do, and it usually allows us time for what I have dubbed "family fun time" where we play a game or read together, or whatever they would like to do that does NOT include a screen :)
As for the schedules for the other kids, I don't know if this is much help, since you have one more than I do, and a baby at that! I would only suggest that you try to get yourself a few hours alone each week - even if it is just a short time for you to go over your calendar and make sure you are organized. I found first grade so hard as a mom, because I was unprepared for all the papers that came home (suggestions, anyone??) and the various meetings and activities I suddenly had to keep track of. I am still working on that, but at least this year I know what will be coming at me, and how important it will be to stay on top of it.
I am still tryingto figure this all out myself. Please keep us posted - I would love to hear all the ideas and what works or others.