Can landlord charge all of us for one person's screw-up?

Yes since the culprit cannot be identified the only thing they can do is to charge all tenants equally.

Maintenance still costs money and there is no need to hire a plumber if maintenance can do it, but they still get paid wages, so the cost is still there to do it and yes they can charge all 6 people ....they have warned you all in writing that they will charge if it happens again

No they can not charge the entire units a $50.00 fee for the dumping of these items. But they can locate the culprit easy enough.

If you're on a month-to-month agreement, they can make any changes they want as long as they give state-required advance notice. You have the right to reject the change, give your notice that you will be vacating, and be out by the effective date of the change. State law must be followed.

If you have a lease, read the lease to see what it says about things like this. There may be a clause that says tenants are responsible for plumbing problems that are caused by tenants (like in this case), but it must be proven which tenant caused the problem - they cannot unilaterally blame everyone and assess everyone.

If it looks like this is the case and you have a lease, write them a letter, make a copy, send the original with proof of delivery. Let them know that if they assess a charge against you without proof that you caused the problem, you will take them to court and ask for punitive damages as well as court costs.

yes they can

They'll probably go after the women with a baby, because most childless men don't flush baby wipes and pads anywhere.

Depends on your lease.