Regarding frequency, yes I was referring to your “Logic Input P” signal, which you stated was to come from a function generator. For a single pulse, this is not relevant. I would, however, use a much longer pulse initially, to see if that affects your results. Based on your o-scope image, I would expect the output voltage to still drop to zero after 10us, but it would be worth checking.
For your simple circuit using only the MAX320, if everything is connected correctly and the MAX320 has not been damaged, it should work. The output you are showing is more representative of a capacitively coupled signal rather than a resistively coupled signal, represented by the circuit below, except that I would expect it to have more of an exponential decay look to it.
If this capacitor were placed on the output of COM2, it would give a similar result to what you are seeing. In a capacitively coupled circuit, the voltage across R would initially pull up to +Vwrite and then decay to zero as the capacitor discharged. If your MAX320 were damaged, the switch might behave in such a way by only capacitively coupling to the output. I would suggest trying Switch 1 or another MAX320 if you have one available to see if you get the same result.
As I mentioned before, if you have ever applied a voltage to any of the pins before V+ and V- were powered, that could damage the part.
Regarding the +Vwrite analog input voltage, when you say “source meter”, I assume you mean a DC power supply with a display to set the output voltage. If so, its source impedance would almost certainly be just fine. However, be very sure that V+, V-, and all of your other inputs and outputs are referenced to the same ground potential. You should have a ground point somewhere on your board (preferably very close to the MAX320 to minimize ground loop size) and the ground lead of each signal; V+, V-, +Vwrite, Logic Input Pulse, and DUT should be connected to that same ground point to ensure that all of your voltages are referenced to the same ground potential.
Regarding possibly using alternative parts, other than replacing the MAX321 with the MAX322, I think the MAX32x series should work. There may be a single IC that can do the whole job, but I have not looked into that yet.
You can certainly solder these through-hole parts into a circuit board as long as you have a board with the proper holes drilled. One can typically maintain signal integrity to higher frequencies using surface mount parts on a well laid out board, but one can make a very functional board with through hole parts, too.
For the oscilloscope, using 1M-Ohm test leads should work just fine. Just be sure to connect the ground lead to that same ground point as I mentioned for all of the voltage sources.