You do not say which size cutting deck you have. I will be writing about the 48", 3 blade cutting deck
The belt on this deck is self tensioned. The tension is provided by a spring attached to a moveable pulley on the deck. When the belt is in place you can manually force this pulley forward/backward and it will release the tension.
If the belt is off track, place the belt back around all the pulleys in the correct path, lastly force this spring loaded pulley forward/backward (depending on your deck, the direction is not important, the tension is) to get the belt around the pulley. If the belt is placed correctly, when you release the spring loaded pulley it will properly tension the belt.
Typically, these belts should not be jumping off. If yours is coming off frequently the belt may be worn. When they wear they get quite a bit longer, then the tension pulley cannot apply the correct tension. Rarely does the tension spring have to be replaced.
The other problem that causes belt jumping is worn spindles and stationary pulleys. The spindles are the most likely to need replacement. You can buy excellent replacements for JD spindles on Amazon for very reasonable prices. The following spindles are super replacement spindles for the 48" deck. They may very well be the same for smaller and larger decks; I have not researched that. Just search Amazon for Silver Streak # 285851.
Silver Streak # 285851 Spindle Assembly for JOHN DEERE GY21098, JOHN DEERE GY20962,
Spindle Assembly / John Deere GY21098 Pack Size:1 JOHN DEERE GY21098, JOHN DEERE GY20962, JOHN DEERE GY20867, JOHN DEERE GY20454 OREGON 82-359, ROTARY 12910, ROTARY 11964 JOHN DEERE D100, D110, D130, D140,D150,D160, LA100, LA105, LA110, LA115, LA120, LA125, LA130,LA135, LA140, LA145, LA155, LA165, X110, X120, X140 Includes grease Zerk, pulley lock nut, blade lock nut, and washer , Height: 7 ", OD: 5 "
I couldn't with any certainty tell you there are no reed valves in the engine of your mower but reed valves tend to be fitted almost exclusively to high performance 2-stroke engines. Compressors have them and it could be argued diaphragm type carburettors have them (with a stretch of imagination).
Most ordinary duty 2-stroke engines use piston controlled ports while virtually all 4-stroke engines made in the last century use poppet valves and I strongly suspect that applies to your mower.
Depends on the type of pump - inline or distributor type or individual pumping elements?
In the absence of accurate marks there are various methods that could be employed but all will need manufacturer's data to set the engine to either tdc or to injection point and then to proceed to time the pump to the engine by a method known as "spill timing" for the inline pump or the distributor type of which there are 3 main types, the commonest having a hydraulic governor, uses an adaptor and dial gauge.
I am guessing the ignition is a self-contained module so unless the kill switch or wire is damaged and grounding or the flywheel magnet has lost magnetism the ignition module will have to be replaced - unless the spark plug connector is a suppressor type and then it is possible the internal resistor is open circuit.
Seems Deere didn't put diagrams on their 100 series mowers, just to save a buck I guess?
Sure are a lot of complaints online over it for this to just be a mistake? Not on the deck like most mowers are, not under hood nor under its seat. Simply not on this damn mower.. I got the belt on ok because I've worked on mowers many yrs but some folks may really struggle to do so, what a joke JD is today.
By turning over for a little while without ignition, do you mean it isn't starting or you are cranking the engine with the ignition switched off?
Regardless of this it seems fairly clear something is wrong with the starter and/or maybe the battery (it isn't unknown for a new battery to be faulty), alternatively there could be a supply problem - dirty terminals, dodgy battery leads, solenoid.
Certainly starter motors become tired over time and need servicing.
Equally certainly, if an engine is going to start, it should take only a few seconds of cranking.
You have probably wasted money throwing new parts at the problem but nowhere have you said you have checked the spark and it is ok or that the spark plug is becoming wetted with petrol...
To start and run an engine needs...
A good cranking speed.
The right amount of fuel being delivered into the cylinder.
A good spark at the spark plug at the right time.
When an engine doesn't start one or more of those requirements is absent and we can't able to guess which. Only systematic checking and testing will reveal the culprit. Check, don't guess!