Plumbing utility sink into washing machine with no vent

So I currently have a washing machine that has its hose running into a 2" standpipe that is coming out of my basement floor. A humidifier hose and AC hose also squeezes into that standpipe and it's pretty tight and janky looking. I'm trying to clean everything up with proper piping, install a laundry box and also a utility sink.

There is no vent or additional plumbing nearby that is easily accessible.

The standpipe is on the left, machine in the middle and the sink would be on the right. Can I have a single PVC pipe that connects the sink to the machine that then connects to the standpipe? Do I need a mechanical vent for it to work? It all seemed to work fine before but I am sure it wasn't to code (Toronto). What is the best way to connect the humidifier and AC to the same pipe? Should I have individual connections piped to the standpipe?

Here's the standpipe on the left with humidifier and washer draining into it:

Diagram of the current setup:

Proposed setup – sink on right and washer on left, draining into standpipe.

Or here's another proposal. My understanding is I need to swap the sink wye and put it above the washer so there's no chance of backing up the utility sink.

What considerations should I be making with regards to venting, angles and heights? Any other considerations?

Thank you for your help!