I share custody 50/50 with DDs dad and she spends every other week with him so we have to be extra organized. Fortunately we get on well so it has never been a problem so far. We have a shared online calendar for DDs activities - her extra curricular are set (Spanish, dance, drama, swimming) and blocked out.
Whichever parent she is with, whenever a letter/invite etc is received, we take a picture, add it to WhatsApp (so we both have the original with all the info) and put it into the calendar. Who ever she is with when the event happens is responsible for making sure she has the stuff required - eg if World Book day falls in my week then I have to sort the costume but if in ExDHs week then he sorts it etc. Invites to parties etc depend on each parent's schedule, whoever receives the invite might not be the parent DD is with on the party date so we just put them all in the calendar and the parent DD is with on the party date is the one who RSVPs.
Every Sunday morning before my weeks with DD I go through the joint calendar/WhatsApp pics and make sure everything is up to date for the week ahead in my diary since I work better on paper and I have everything I need (or at least have it ordered on Amazon Prime). We have a whiteboard in the hall and I update it for the coming week showing all our appointments/classes/work hours/ parties etc.
As for stuff - I have a Kallax 4x3 unit in the hallway. One column is mine, one is DDs, one is shared and one is for DDs extra curricular classes (there are boxes for Spanish workbooks/dance and drama uniform/Swim stuff and anything needing washing gets washed and returned to the correct box)
The bottom cube is for school/work shoes (3rd one i keep a box with hats/scarves/gloves in winter and sunhats/sunscreen and sunglasses in summer); the middle cube is where any material we need for the following day (eg pe kit/swimming things for DD, gym kit/library books for me) is put the night before; the top cube is where DDs school bag lives and my work bag.
One of the shared cubes is where we have an in tray for mail and where DD knows to put any letters/invites as soon as we get home. I deal with the mail in the evening (I spend 30 minutes after DD goes to bed doing household admin/tidying) so if it isn't in the in tray before bedtime then tough luck because I am not signing permission slips as we are rushing to get out in the morning. The other one has the kallax drawer inserts - the top drawer is full of random bits and bobs we always need on the way out (pens, spare keys, name labels etc) and the other drawer has envelopes, small notecards and a small pot with a few 50ps/£1/£2 coins so I always know I have change on hand for school money requests.
When DD comes home she hangs up her coat, puts her shoes in her cube and takes out any letters. She then takes her school bag to her room to do her homework while I make dinner, and when she is finished she brings her school bag and anything she needs for the following day and puts them in the cubes. That way I can see at a glance if anything is missing as the cube is empty, and everything is ready to just grab and go the following morning. It sounds very rigid when written down but I find that the routine works well for DD ashe always knows where her things are as they all have a specific place they live.