"My Company Tickets" -- everyone who needs them should
have the same CustomerId.
For instance: Everyone in a department has the same CustomerId. Multi-department managers have a CustomerIds of depta,deptb,deptc.
Done.
With regard to the conflicting notification systems, you did that to yourself. Or, more accurately, your corporate policy did that to you.
"We need the following behaviour: when one agent replies something to customer
in a ticket via Web-interface - that is, 'SendAnswer' via 'AgentTicketCompose'
page - then all agents subscribed to this queue must receive mail notification
with that text"
Really? That's not OTRS fault. Who does that? You might as well yell at Microsoft because distribution group members don't receive emails sent by one of the distribution group member to someone who's not a distribution group member. Speaking of -- that is the solution to your problem. Create an email distribution group for each of the queues and cc or bcc that queue's distribution group when sending the Answer.
It won't stop doubling, though. Turn off notifications in agent preferences.