Bandwidth of an oscilloscope with differential probe
From experience, can anyone tell if 7 ns timing difference between
  signals can be captured if the rise time of the scope is 5 ns vs 1.75
  ns?


For a signal with a 7 ns rise time a 5ns scope would see 1.4pts during that time
The same signal with a scope that samples 1.75ns would see 4pts.
How many points to do you need to see?

Question 2 is a matter of opinion, it's a trade-off of money vs need.


  I will be using a differential probe and most of the differential
  probes that I came across are 25-50 MHz bandwidth range. Could adding
  a differential probe with BW of 25 MHz affect the above timing
  measurements given that the signal frequencies are going to be less
  than 10 MHz?


A 70Mhz scope with a 25Mhz probe will only be able to 'see' signals at 25Mhz. For example, if you had a 50Mhz sine wave with a 25Mhz probe, it would probably be so attenuated you wouldn't be able to tell it's there. The oscilloscope would be more than happy to display points at 70Mhz. So you are limited by the probe. A slower probe is like having a filter on the scope.