Just had same problem. Did the same thing, bought a solenoid that fixed nothing. Also jumped from battery to starter and got spark as you did.
Battery terminals were clean BUT the positive cable attached to battery was rusted. Thoroughly cleaned cable end and bingo, it starts.
The deck has 7 positions. The lowest sets the deck on the concrete. Levels 1 thru 5 are basically too low to cut grass unless you are cutting greens or a fairway. Level 6 is boarder line too low for the average lawn. The highest level is ok.
If there is excessively high grass there is no room...