What is a good process to replace a discontinued part?
I'll tackle point 1 with your specific use case in mind: a solution could be to design a small daughterboard on which to fit the new part with possible ancillary components. That little board would have two strips of pin headers dimensioned in the right way to be plugged in the socket of the old part. If the old part is not socketed, careful desoldering would be done and the header from the daughterboard soldered in their place.

Of course this assumes that:


there is enough space once the old part is removed; but in your case since the new part is SMD you have some headroom, especially if the daugherboard has 4 layers, so that routing the new part pins to connect to the old part's holes would be easier.
parasitics are not a problem because you keep the operating frequency sufficiently low. The daughterboard will introduce more stray capacitance and inductance relative to a setup where the part is soldered in the right place.