Yes, safe to say needle and seat (or at least...the needle) should be replaced.

"Handy with tools and follow instructions"
Disconnect the small wire you will find on the starter solenoid (this WILL NOT be on the starter). With the small wire not connected to the solenoid, try to place the battery cable back on the battery (but first read my special instructions at the end of this post). IF the starter does not engage now, very carefully just touch the small wire back to the terminal on the solenoid. If the starter engages when the small wire is in contact with its terminal, the key switch has come apart inside and is shorting to the start position.
BUT.....going backward a step, IF the starter still tries to engage when you touch the battery cable to the battery post while the small wire is disconnected from the solenoid, the solenoid itself has become stuck and needs to be replaced.
Proceed cautiously, batteries can explode. If you disconnected the battery positive cable (you did not say which) you took a risk of arcing the tool against a grounded surface. You would be safer if you remove the battery negative cable from the battery......then reconnect the battery positive cable to the battery while the battery negative cable is disconnected. To safely perform the tests I outlined above, you need just the negative cable of some booster cables (like for a jump start).
Connect one end of the booster cable to the battery and lay the other end where it can't touch any part of the tractor (or just hold it in your hand). When you get ready to do the tests, make your "battery connection by touching the remaining end of the booster cable to the loose batter negative cable of the tractor. Do this while holding the tractor battery cable as far from the battery as it will allow you to. This way any sparks (and there may be some) will be far away from the battery, reducing the chance of explosion.
But you still should wear some eye protection, that way if the battery explodes, you can still see how to find the water hose or shower to wash the acid off.

Thank you! I will do this test and take all the caution measures you outlined.
To tell you the truth, I disconnected the positive,(I was on that side and it was closer to me) but did it by hand since it's held by a "butterfly" screw which is easily removed...
Question, are these parts generic or I have to order from the OEM supplier?

The starter solenoid is absolutely "generic" and can be replaced by a great many different solenoids (match up by sizes of terminals, number of terminals, and configuration of terminals).
The key switches are "fairly generic", but you must pay close attention (very close attention to details) to the identification letters and their arrangement on the back of the switch. If there is a single molded plastic connector body that plugs onto the switch, the above advice is extremely important.
If the tractor wiring connects to the switch via individual wire terminals, the arrangement of the terminals is less important on the replacement switch, BUT... labeling the wires with tape or tags according to which terminal they are on before removing them (use the letter ID of terminal: eg S, M, B, etc.).
Other things to watch out for when choosing a replacement switch is the way the switch mounts to the tractor include the following:
The diameter of the mounting hole, the shape of the hole... some holes are "D shaped", others may have one or two notches to accept alignment lugs on the switch body.
Also, I always try to remind people to use their digital camera as a tool to record images of just exactly how a component is mounted BEFORE removing anything.
Then you can refer to the images later when you re-assemble it all.

Mownie, thanks for your fast response and concise explanations.
I'm looking at the diagram now, and there are what seems to be two "small wires" coming out of the solenoid. One is marked as "red" going to the ignition switch and the stator, and the other marked as "orange" going directly to the ignition switch passing thru the clutch brake switch.
Which one do I remove? I'm thinking the orange but not sure.

Look at the actual solenoid on the tractor.
If the diagram you are looking at is accurate(??) the orange wire would be the one to disconnect for my test.
The small red wire referenced in the diagram would be the main "hot feed" circuit that supplies battery voltage to the ignition switch, and the small red wire circuit should have the main fuse in it somewhere. The small red wire you refer to should be attached to one of the big terminals on the solenoid along with a battery cable, specifically the battery positive cable.

Yes! It was the orange, the one attached to a little spade on the base of the solenoid. Did the test and the solenoid seems to be the culprit, the starter would kick in even with the wire disconnected.
Took it off and I'll see if any of my local shops has one. I'll update with the results.

Back to my issue #1; I got a carb overhaul kit yesterday, I'll take teh carb apart again and change everything. One thing I noticed though, was that the seat is brass, not rubber, would it be absolutely necessary to replace it? Or just the needle is the one tha usually wears out? (I see it has a rubber tip). If so, I think I need a special tool to retrieve it right?
Also, the float didn't come with the kit, and they didn't have it in stock, would it be a good idea to order one and replace it too?
Thanks again for your help.

NOTE: If the needle valve has the rubber tip-that will act as the "seat", and there is no soft seat in the carb, just the brass part you see in there. As for the float, try this: Put an inch or two in a coffee can, and set the float in it for an overnight set. If the float is bad, it will not be a float-it will be on its way to being a "sink"!
If that is the case, a new one is usually the answer. They are usually available to buy separately. Also, the needle may come out with the float. But, i'd leave the brass seat in place. by: RJ

The brass seat is a fairly durable item. If the seat shows no signs of pitting from corrosion or other damage, it should be alright with a new needle.
Floats are reusable unless there is physical damage.
Floats made of brass (rarely found any more) can develop a leak permitting gasoline to enter. This renders the float useless.
Floats made of composite foam are not susceptible to leakage and may be reused if the float is intact (good and solid with no sections missing). Floats and bowls should always be inspected for signs of rubbing against each other.
This would be apparent by seeing a scuff or brush mark on the metal surface of the bowl. I doubt you will find any rub marks on your carb.
Your carb most likely suffered from formation of varnish from degraded gasoline during its long period of "storage".

Cool! With "an inch or two" you mean of gas or coffee? (I know it sounds like a funny question)

So, are you saying your float is made of brass?
If it is brass, there is a quicker and more certain way to test it than the coffee can method.
If your float IS NOT made of brass, no need for any "float tests" other than a visual to confirm it is intact.

No, what is brass is the seat. The float is plastic.

***"The float is plastic."***
No need to test this float for leaks.

Ok guys, thanks a lot for all the help.

Update on Issue 1: Put the new needle, inspected float and was ok, put all new gaskets, and added a gas shut-off valve. Changed the oil since it had gas in it. She's running great, had it iddling for like 10 minutes, reved it up to full throttle and did this several times, no signs of flood or anything else!
Update Issue 2, Well, I found a replacement solenoid locally for $14.99, and worked fine. Switch seems to be fine and the problem is gone.

Now on to the next phase, blade and belts. I might have more questions about that once I start tinkering around!

Thanks again!!

We like reading success stories. Thanks for updating with good results.
Tinker away!

Well, I came home to a little gas puddle under the mower. It looks like it's coming from the screw that holds the tank, it's very small, but it's there. This is the first time I leave gas in the tank overnight, because I have been draininng it because of the flooding. Any idea how to repair this?

Just to follow up, I went and tried one of those "Plastic gas tank repair kits" from my FLAPS (Frienly, Local Auto Parts Store)which is not more than a epoxy compound in a roll form. Had to apply two thick coats to completely seal the leak but the tank is full since saturday and no leaking so far.
Took the blade off yesterday and sharpened it, inspected the belts, they look OK to my eyes, there's no noises or anything when they engage so I'll leave them there for the time being.
Engine still purring like a kitten, stars at first try.

Mower is officially ready for the Field Test.