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

**P1**. 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:

**P2**. 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.

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, or more than one point
on one $C_i$.
The efficacy of this plan depends on the degree to which
P2 is completely solved in its various guises.

**References**

1.
M. J. Nielsen. "Triangles inscribed in simple closed curves,"
*Geometriae Dedicata* **43**: 291-297 (1992).

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

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

exactnumber? (The usual definition of the word "algorithm" is modelled after our familiar digital computers that don't have infinite-precision real numbers.) – Sergei Ivanov Sep 17 '10 at 20:39