What would be a good "jack-transport aware sequencer app" to use with
the MIDI drumset (which has sound module) and the MIDI
keyboard/controller (which I think depends on software for the variety
of sounds)?

For drums Hydrogen is always a popular option - try and get the latest release if at all possible.

...I mean something other than PianoTeq. (No offense to anyone
advertising that but I cannot spend that $$$ right now.) :-\

Your other options for a piano sound are, a giga sample file + linux sampler or sound font + qsynth. If I remember correctly somebody posted a piano soundfont here some time in the last year or so (search the archives).

For more synthy sounds I would take a good look at Yoshimi which is a version of Zynaddsubfx which is being made more realtime friendly.