As for your second and third question, the following may help:
useCustomSpan = {
\override TextSpanner.bound-details.left.text = \markup{ \musicglyph #"pedal.*" }
\override TextSpanner.bound-details.left.stencil-align-dir-y = #0
\override TextSpanner.dash-fraction = 0.5
\override TextSpanner.dash-period = 1
% \override TextSpanner.thickness = 2
}
\score {
\relative c'' {
\useCustomSpan
c4\startTextSpan d e f g2\stopTextSpan e
}
}

You can modify the dash-fraction
and the dash-period
properties of the TextSpanner
object in order to get the desired dash pattern, and you can change the stencil-align-dir-y
property for the left text to #0
, which means that is is vertically centered.
As for your first question, there is also the property TextSpanner.thickness
. However, I think the line thickness is already quite alright, so I commented it out in the above code.