I have a pentium 4 PC running Win ME, and an HP Laserjet 1200 printer
connected via parallel cable. My question is in two parts:
a) When I have several applications running, the system fails to send
info to the printer upon the Print command. It will do nothing until I
close some or all of the applications. What's going on with that, and
is there a fix?
b) The data transmission to the printer is depressingly slow for large
files. Would the USB connection be faster thatn the parallel? Is there
an upgrade for the printer to USB 2?

According to my research the HP Laserjet 1200 printer does support USB
2.0.  Provided that your computer has USB 2.0 I would recommend using
that interface over the parallel interface as it is much faster.

Your slow printing problems may be related to the mode that your
parallel port is set to in your BIOS.  In most BIOS's there are 3
different settings for the parallel port.  ECP, EPP, and
Bi-Directional.  From the information I gathered, your printer is
meant for Bi-Directional transfer, so you may want to try fiddling
around in your BIOS to change this setting.  Another possible cause
could be poor driver support in Windows ME, but I can't say for sure. 
Check HP's support website for their latest drivers.