The reason for sluggish operation of the automatic trip, and inconsistent or no auto function in this chassis is usually sticking auto trip levers in the main cam wheel. The levers should be absolutely free of oil or lubricant; as DSJR says, they should rattle freely in the cam gear. Over time, oil/grease residue gathers on the parts, dries up and causes drag on the levers. The spring, if undamaged, is very light but should be sufficient to push the associated levers and the auto trip levers.
Happily, your Stop Lever is okay and not stuck. Garrard greased them at the factory and the grease would dry up over the years. Yours may have been serviced previously and a modern, more stable grease applied.
Speed of cueing drop can be adjusted, but only slightly, by moving that spring you describe. Rarely, changing the damping fluid can improve cueing action but unlike the cueing on many other turntables (Duals after the 1209, many Japanese models, some other Garrards, BIC are examples) the SL95's cueing damping fluid does not leak out!
You might get temporary improvement in idler traction with a rubber treatment based on oil of wintergreen or bu rubbing a solution of rosin in solvent (liquid solder flux, for instance) into the rubber and wiping off the excess so that rosin only remains in the pores of the rubber. But the only permanent solutions are a rebuild, new rubber on the existing idler; or a new old stock idler, though the rubber may have oxidized in certain climates, becoming slippery. Original Garrard idlers would be about 40 years old.