pedal..is this the Tuesday WPO game?

Axis prob wont appreciate me helping you but who cares, I'm not in it.
Ok to update my current situation. I'm the USSR and need to currently maintain garrisons/defense in Syria, Suez, Persia, and Indian borders. So, unfortunately I need to keep at least some substantial forces in those regions.
I don't think all of those are desert. it may be better to leave the desert empty and let the enemy take it. Then when he does you attack him on the desert, and he loses troops both to attrition and battle. Like a trap. This would not apply to suez, as once that is taken you lose access/gain access to the med depending on gibralters ownership.
My actually TC load is around 528/498 - so I should be able to ween that down. Does it make that huge of a difference to be over by so little? And reversed, does the lower down I am IMPROVE anything or is it simply a matter of 'all or nothing'.
I believe there is an immediate severe penalty once you go over 100%. Then the penalty gets linearly worse as you go up. So the moral us, get it at 498/498 and you will notice immediate benefit. Lower than 100% doesnt mean anything.
Do the hospital techs do anything/make any substantial difference?
They help your trickleback modifier, which means that troops that leave from attrition come back right away. I think they help something else too but cant remember. I've never done testing on the degree of help, but I always research the tech if its same year or in the past. You are SU though...MP should not be an issue so it may not be a big of a deal to you as say UK or Germany.
Ok - how do I improve my supply lines? Or do they auto improve as my TC does?
Supply lines are a direct equation of quality of the province in the supply line * TC modifier. The supply line should be chosen from your capital or resource drop to the destination, and the computer should intelligently pick which provinces have better ESE in between these two locations. That being said, you cannot control the quality of the province aka terrain type and infra level in the short term. What you can control is TC, which you can lower with all the reasons I gave you before.