The 7805
    is a
    non-adjustable
    current
    regulator
    and it just
    happens to
    be set to
    34mA?

No, it’s a 5V voltage
regulator. (The 05 in
7805 is for 5V) It
maintains a constant
5V between it’s
output and ground
terminal. In this
circuit there is a 150
ohm resistor between
the O/P and gnd
terminal so the 7805
puts a constant
voltage of 5V across
the 150 ohm which
results in a constant
current of 5/150 or
~34mA through the
7805. The I/P of the
7805 is connected to
the cathode of the
tube so the current
through the 7805 is
the tube bias current. 

At switch on the tube
will be off. The 7805
will turn hard on to
try and put 5V across
the 150 ohm so the
7805 I/P (and tube
cathode) will be 1V
or so above ground
potential i.e. there
will only be ~-1V
between grid and
cathode. As the tube
warms up the current
will rise until the
7805 is able to put
5V across the 150
ohm then the 7805
will start to turn off
and the voltage
across it will
increase until the
voltage on it’s input
(and tube cathode) is
equal to the bias
voltage the tube
requires to maintain a
current of 34mA. 

The input voltage to
the 7805 therefore
depends on the tube.
It will be about 12V
for a EL84 but could
be over 40V for a
6L6 so you would
need to use a
regulator with a
higher input voltage
rating or you could
put a zener in series
with the 7805. 

Dave