Interpolate/Decimate with a single filter?
Yes, you can interpolate and decimate at the same time.  This is called "resampling".  If you google resampling you will find lots of information about it.  And yes, your reasoning about resampling is mostly correct.

When thinking about resampling theoretically you usually put the interpolation first to avoid Nyquist issues.  An interpolator is upsampling followed by a low-pass filter to get rid of the aliases.  A decimator is a low-pass filter followed by dropping samples.  You can combine the two by combining the two low-pass filters into one filter.  If the overall rate change is greater than one, the low-pass filter has the same pass-band/cutoff as the interpolation filter.  If the overall rate change is less than one then it has the same pass-band/cutoff as the decimation filter.  The filter operates at the interpolation sample rate.

You can make the filter more efficient by not actually upsampling the data or dropping filter outputs.  Instead, you use a polyphase filter for the interpolation and simply not calculate the results that will be dropped, but that is a topic for another day and another question.