A technique to measure the decay characteristics is given in my answer to "least square fitting to inverse exponential function". This method also gives you a fitness metric which can be calculated in a single loop as well. Note my advice in the comments to use a +++--- pattern instead of a +-+-+- pattern that I give in the answer when finding the rate of decay.
A good amplitude is simply your max value. You may want to average the peak and it adjacent neighbors instead because the peak is sensitive to sampling timing.
A good characteristic duration would be the distance between your nearly vertical line on the left and you fitted decay curve on the right at half your peak height.
An aspect metric could be formed by dividing the width by the height.
That should give you a good set of numbers to identify and describe your events. They are fairly efficient to calculate as well.
The leading spike looks like an event to me as well. If events of interest have to have a certain height it makes this problem that much easier.
Hope this helps.
On second thought, you are probably better off measuring the width of your pulse at a fixed height. Perhaps, your cutoff threshold would be a good height because you would always get a value. For the left side you should to a linear regression to find the best line.