A B767-300 EICAS alerted HYD QTY C followed by HYD SYS PRESS C at 10;000 FT after takeoff. The QRH procedure was completed; an emergency was declared and the flight diverted to a nearby airport; where during approach; the TRAILING EDGE DISAGREE light illuminated while flaps were extended using the ALT FLAP SWITCH as per the QRH.

Narrative

Just after takeoff departing and passing approx 10;000 FT; 250 KTS; clean configuration. EICAS warning; HYD QTY (QRH abnormal) followed very shortly (approx 30 seconds) by HYD SYS PRESS (C only) 767 (QRH EMER). I pressed the status page to see the volume of the Center Hydraulic System reach zero (with RF). I grabbed the QRH and Captain made a radio call to ARTCC declaring Emergency and gave them notice of our intent to divert. We started divert and started the QRH procedures in order above. No action items for HYD QTY. Started HYD SYS PRESS QRH as we diverted. Captain focused on flying the plane and I ran the radios (I speak some of the local language and this helped with communications) and did the QRH procedures. We completed the QRH and continued our divert while we reviewed items lost. We reached VMC conditions and had the divert airport in sight when we started the ALT FLAP extension portion of the QRH procedure. Normal operations on checklist until we attempted to select Flaps 5 - we received a Trailing Edge Disagree light and EICAS warning. We initiated SATCOM link with Dispatch and asked for help from Maintenance as we advised our Dispatch of situation. We spoke to a Maintenance Technician that told us to put the ALT Flap from 1 back to zero and we declined to follow that advice. We asked Dispatch to speak with the Chief Pilot. Captain ran through checklists until we were all together. At which point all 3 of us confirmed we were a multiple system failure. We then started to plan the NO FLAP/NO SLAT LANDING and the FLAPS Landing other than Flaps 30 procedures. We finished the HYD SYS PRESS (C only) checklist and extended the gear. We computed an approach reference of 210 KTS and a touchdown speed of 190 KTS. We intercepted the final approach and shot a VOR XX to a visual approach when Captain flew a very stable approach to a very nice landing. We touched down and came to a stop with approx 1;000 FT remaining; using AUTO BRAKES 3 (stayed armed all the way to stop) and reverse. We asked to be towed to parking and after the fire marshal inspected us; we proceed to the ramp area under tow. After maintenance repaired the plane we flew home on and had a Rudder Ratio EICAS warning enroute. We would like to make the following [recommendations]: 1. All of our carriers airplanes that fly beyond VHF radio range need SATCOMs; this was a critical 'lifeline' for us. 2. We need a nearest airport button on the face of the FMS; in all fleets; so when diverts occur we can get data into the box fast. 3. On five different occasions this Captain and I had EICAS QRH procedures to follow. Objectively; this shows a significant increase in emergency actions; certainly the board can draw conclusions from these facts. When I ask maintenance they say they are 'cut to the bone'.