Permit me to reformulate a specific version of Q3 that
Ellipsissi posed in the comments:

> <b>P1</b>. Given three non-intersecting circles
$\{C_1,C_2,C_3\}$,
find all triples
$\{p_1,p_2,p_3\}$ with $p_i \in C_i$
such that $\triangle p_1 p_2 p_3$ is
similar to a given triangle $T$.

This differs from the posed question in (a) the non-intersecting condition,
and (b) not demanding that a specific angle be realized at
a specific corner $p_i$.
The form above is analogous to this problem:

> <b>P2</b>. Given a plane curve $\gamma$,
find all triples
$\{p_1,p_2,p_3\}$ of points on $\gamma$
such that $\triangle p_1 p_2 p_3$ is
similar to a given triangle $T$.

Much is known about P2, under various restrictions on $\gamma$.
For example, if $\gamma$ is a smooth Jordan curve,
then I believe it is almost completely understood now,
through recent work of
Benjamin Matschke, and of
Jason Cantarella, Elizabeth Denne, and John McCleary.
See especially Cantarella's fascinating [web pages on the topic][1].


So, here is a high-level plan for P1. Connect the three circles by thin
corridors to form a plane curve $\gamma$. 
Solve P2, and discard solutions with points on the corridors.
The efficacy of this plan depends on the degree to which
P2 is completely solved in its various guises.


<b>References</b>

1.
M. J. Nielsen. "Triangles inscribed in simple closed curves," 
_Geometriae Dedicata_ <b>43</b>: 291-297 (1992).

2.
Benjamin Matschke.
"On the Square Peg Problem and some Relatives."
[arXiv][2] (2009)


3.
Wikipedia article on the [Inscribed Square Problem][3], with triangles discussed under "Variants."


  [1]: http://www.jasoncantarella.com/webpage/index.php?title=Square_Peg_problem#Results_on_Inscribed_Triangles
  [2]: http://arxiv.org/abs/1001.0186
  [3]: http://en.wikipedia.org/wiki/Inscribed_square_problem