Op-Amp suggstions for pulse trains?
Use a comparator and if you want really fast edges try the MAX999. It's output voltage will not quite be 5V peak-to-peak but more like 4V peak-to-peak but it's quite suitable for driving a logic input. 

The MAX999 has 2.3ns rise and fall times.

Should you require a precise 5Vp-p output I'd still recommend a comparator. 

It should be noted that amplifying a signal by (say) 10 does not "sharpen the rising falling edges of the pulses" by 10. If the rise time of the original pulse was 1ms then after amplification the rise time will still be 1ms. If you want to sharpen edges then use a comparator - if you want to preserve amplitude information don't use a comparator - you can't have both.