2001 taures ses...sputters and idles "hard", "rough" and stalls. once we get over 30 it runs fine

Vehicle information needed . How many miles ? Any recent work on the engine ?

Fuel pump

check timing, adjust valve clearance, check vacuum leaks. these are some that need attention.

Disconnecting the battery sometimes corrupts the CPU program and it takes the use of a scanner to reprogram it . What you have done sometimes fixes the fault but like all CPU programs it only takes one bit to throw out the program.

Check you tps sensor sure that's your problem

. The answer is to replace all the spark plugs and wires. Even if the plug appears fine to the ***** eye, and the wire has no visible damage, there could be problems with it that cause it not to fire properly. if you have coils on it, check them also.
An ignition coil that is failing will still provide current to the plugs, but it might not be enough for a full spark and the car will run sluggishly. If the current being provided by the coil isn't consistent, the car will run rough. make sure your getting good spark from coil first. good-day !

Although is could be a fuel filter, it could also be a fouled spark plug, or a bad spark plug wire. Once you've ruled these things out, the problem could be the TPS (throttle position sensor). This controls your idling speed under all conditions. A faulty TPS will usually idle fine when the engine is cold but once warm, will tend to sputter and in some cases, stall.

My 1990 S10 pick-up had a similar problem too. It would start up fine (while the engine was cold) then once the engine warmed up it would idle very rough, sputter, and even stall out. My grandpa and I found what was wrong finally. The fuel distributor had been cracked, so when the engine warmed up it would miss (giving either too much or not enough fuel). So I recommend that you check that. Hope this helps.

check the actual fuel line. Also is it fuel injection or carberator

The CAMshaft position sensor is faulty.