Most likely the "flapper" (spring metal piece) that is 'posta be attached to the
lever/shaft that actuates the release valves has broken or is MIA ( quite common)
you have 2 options:
#1: repair the flapper mechanism or actuator if you prefer, or
#2: replace the 2 release valves, with the newer style that are automatic. and need no activation
On the crud in the tank:
take the tank off, drain and refill with some CLR rust/lime remover and add a handful of 1/4"
or 5/16" nuts or pea gravel size rocks.
and shake the shit out of it.
It might take ya a while shakin' an sloshin' till it is cleaned up inside.
but keep at it, might take some sittin' an restin' then go back at it.
When you are satisfied with the results, wash it out with soapy water a few times and let it air dry in the sun.
Reinstall and use a GRAVITY inline filter just incase your favorite fill can is adding rust/crud to it while filling.