Welcome to the world of Lambretta. Unfortunately you seem to have a number of problems.
Even tough the kick start sticks down does it turn the engine over eg engage with the gearbox. If it does it could be a broken spring, if it doesn't return is it catching on the tail pipe of the exhaust, a lot do look for scrapes on the pipe. if so loose a bit of the metal plate that sticks out the side.
The stalling when you select gear may be your clutch plates sticking. especially if its been standing for a while. You can try putting it in gear engine off, pull the clutch in and push it should free wheel with a bit of resistance but not a lot. I have a bike that did the same put the rear brake on and give it just a few revs as you drop it into gear it should kick free. (don't have it screaming its head off or you will be down the road so take care)
you also mentioned a puddle was it fuel or oil? if the gearbox is empty it could cause your clutch problem.
Stickys book is a god send as is joining a club. learn to tinker its half the fun as it can get very costly. Comments about simply putting a new carb on just opens up another can of worms IMO. The float will be sticking sunk or it has a black tip. These are effected by the modern additives to fuel so the leak. the floats can crack do the don't float or the wee pin that the float is mounted on can be a tight fit between the support posts, all easily checked and fixed for a few bob.
I'm a bugger and break all the rules never turn my fuel off, I find it embarrassing when I go screaming off and run out off fuel after a few hundred yards

so its always on, but not bad advice
Good Luck