Starter/valve problem with MTD lawn tractor?

The valves are doing their job- providing compression. The problem appears to be a weak starting system. Check the voltage in your battery. Jump it with a known good battery- if it cranks right up, it's your battery. If not then T/S the starter system. Clean & inspect ALL your starter cables. Let us know what you find.

In my experience starters don't get get weak, they either work or they don't. I would check for bad connections at the ground cable, solenoid, and starter. If any of these connections get hot when trying to start, you should clean it thoroughly. Also check the battery you bought to be sure it is at least 220-230 cold cranking amps (cca), 260 would be better. A lot of the $19 walmart batteries are 140-160 cca and just don't have enough juice.

Valve problems would cause less compression, not more.

A build up of carbon can cause increased compression. Usually this is only a problem on poorly maintained machines. Constant running with a dirty or partially clogged air filter will cause carbon build up over a period of time. Cleaning it is as simple as removing the cylinder head scraping the carbon and putting it back on. Always use a new head gasket and torque the bolts to 12-14 ft-lbs. Removing excessive carbon build up will make it easier to start and increase the power output.