Small Engines (Lawn Mowers, etc.): Briggs and Stratton Model Series 407777 Riding Mower, briggs and stratton, v twin

Question
At the bottom of the bowl float there is what I beleive a solenoid with a wire that plugs in?   
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The text above is a follow-up to ...

-----Question-----
I have a B&S 20 HP V-Twin

Recently I replace my battery and put the cables on the wrong terminals, the new battery was opposite the old and I never thought to check.  Well I found out because my starter would spin however wouldn't engage and I noticed it was spinning backwards at which time I checked the battery and noticed that problem.  Well I switch the cables on the battery however the engine will turn over but not start unless I pour gas directly into the carburetor then it will start for a second and die.  Here is what I have checked and wondering if you thing the Solenoid could be bad.  I replace the fuel filter and took the gas line off were it enters the fitting for the carburetor and cranked the engine and watched fuel squirt out the hose so it appears the fuel pump is working.  I took the bowl float off and it appears everything is fine and clean.  I have also tried to blow into the fitting that goes into the carburetor but I can't get air thru however I am not sure that I should be able to.  I did get my compressor out and I could get air and fuel thru this way but again I am not sure that is good or bad.
So my questions are:
1. Do you think my solenoid is bad?  It appears they are around $40 and I do not want to purchase one if there isn't a chance it is bad.  
2. Could hooking up the battery backwards blow a solenoid?
3. Should I be able to blow directly into the carburetor thru the fuel path?

Thanks, please advise

Mark Posner

-----Answer-----
Rarely will damage a solenoid by connecting the cables incorrectly.

Here's what you know so far.  The ignition, timeing and compresseion are not the problem because it starts when primed with fuel.

This leaves the fuel system.  Do you have any wires that connect to the carburetor, such as a fuel shut-off solenoid?

Let me know.
Eric

Answer
Run a jumper wire, I have one with clips on it, from the positive battery cable to the solednoid.  Unplug the wire from the soleniod and clip the other end of the jumper wire to the post on the soleniod.

Be careful not to touch any other parts other than the positive battery terminal and the post on the solenoid.

You should hear a click when you connect the jumper wire to the fuel/carburetor solenid. Try to start the mower.
Let me know what happens.
Eric