Can an operator/wireless phone company lookup your calls or SMS/text messages via your IMEI number?

For example, if a mobile phone company takes your IMEI number, can they see what SIM card (phone number) was used in the phone?

Can the wireless phone company see if an SMS was sent from a phone by looking up your IMEI number? i.e. match the SMS to an IMEI number?

If someone receives an SMS can the wireless phone company see the IMEI number the SMS was sent from?

See More: IMEI (International Mobile Equipment Identity)