Here is a very natural question:

Q: Is it always possible to generate a finite simple group with only $2$ elements?

In all the examples that I can think of the answer is yes.

If the answer is positive, how does one prove it? Is it possible to prove it without using the
classification of finite simple groups?

The answer to your question is yes. Moreover, if you pick two random elements from a finite simple group, then they generate the whole group with probability which tends to 1 as the size of the group grows. There are even stronger results in this direction, but I am not an expert in the subject so you will have to look for it yourself. You should look for papers by Liebek and Shalev, Lubotzky, Kantor, and there are others who I am not sure about now. 

All of these results require the classification. There are very few results regarding finite simple groups which do not require the classification.

Edit: here is a link to a fairly old survey paper in the notices: http://www.ams.org/notices/200104/fea-shalev.pdf. There are many new developments in the last decade. 
