If they were able to find your name on a particular flight then they must also be able to find the ticket number
You simply have to ask - 'do you have a ticket number against this booking' - if they are not sure what you're talking about then you say 'it begins with 037'. (you are not asking them to 'do' anything
per se)
Seat allocations mean little. Sometimes you change a seat allocation but the original seat still shows on US airways.
That being said - there haven't really been any reported problems with CA bookings. I have had problems with SQ (so has one of my friends). And some people have had problems with TG (but this seems to be around the current renumbering of flights from TG to Thai Smiles - but these are mainly regional flights exBKK).
The main thing you want to do with a ticket number is confirm the booking has been, well, ticketed! With CA's strict 'immediate ticketing' rule, if that ticket number isn't in there then the whole thing may not be good for travel (but unlikely
)