If the car dies after 10 minutes it sounds like the vac hose from the ECU to the inlet manifold sucking itself closed, this will cut the engine out. Go on ebay and get yourself a metre of 4mm silicone hose
http://cgi.ebay.co.uk/4MM-ID-QUALITY-RED-SILICONE-VACUUM-HOSE-TUBING-1-METRE_W0QQitemZ290170998197QQihZ019QQcategoryZ72205QQssPageNameZWDVWQQrdZ1QQcmdZViewItemA metre is enough to replace all 3 hoses on a mk3 16v ABF engine. If you take off the rain scuttle panel in front of the windscreen you'll see where the hose goes from the ECU to the inlet manifold and chances are it'll be soft, weak and crispy. Use the shortest route for the new hose through the bung by the ECU. You don't need to clip the hoses on either, just push on. The other vac hose to replace is from the inlet manifold to the fuel pressure regulator valve. The third is under the PCV valve which will need changing too.
That's the most common fault on them. After that i'd clean the throttle body. Then i'd clean the ISV too. Another one to check is the switch on the throttle body where the accelerator stops. Spray the small switch with WD40 and make sure it moves in and out freely. If you have an induction kit (or not sometimes) then make sure that any parts of the throttle body that have vac hose ports not in use are blocked up. If you have an induction kit then sometimes where the hose from the throttle body to the airbox used to be, the hose is pulled off and the throttle body is not blocked off causing a massive air leak and poor running. VAG COM will not show faults with relays so i'd change the ECU and fuel pump relays too. VAG COM will however show faults with the crank speed sensor so a quick scan will reveal whether or not it needs replacing. VAG COM will also show faults on the coolant temp sensor too which can fail. It's worth replacing though because VAG COM sometimes doesn't pick up faults with them. They might only be broke when reading between certain temperature tolerances. The normal fault is when the engine is hot and the temp sensor is telling the engine it's actually cold. The ECU will push more fuel in because it thinks it's cold which causes over fuelling and poor running. On a mk3 of this age it really needs a hardcore service..
Coolant flushed and topped up with pink ready mixed (NOT WATER!)
Oil flushed and changed
Oil filter changed
Fuel filter changed
Spark plugs changed
Air filter changed (or cleaned if a performance one)
Ignition leads changed
Dizzy cap and rotor arm changed
Coil changed
If your car isn't starting, it's highly unlikely to be the temp sensor causing this. My bets are on the ECU vac hose or an elctric problem with the dizzy/coil/ht leads/rotor arm