The answer to the first question is "yes". This is called conformal gluing, and the proof is based on the following lemma due to Lavrentiev: Let $\phi$ be an increasing diffeomorphism of $[-1,1]$ onto itself. Then there is a simple curve in the unit disk connecting $-1$ and $1$ breaking the disk into two domains $D^+$ (above) and $D^-$ (below), so that there exist conformal maps $f_1$ from the upper half-disk to $D_+$ and $f_2$ from the lower half-disk to $D^-$ which satisfy $f_1\circ\phi(z)=f_2(z)$ for $z\in [-1,1]$.
For a beautiful proof, see Goluzin, Geometric theory of functions of a complex variable, (English translation: AMS 1969), Chapter 11.
The condition that $\phi$ is a diffeomorphism can be substantially relaxed (See Ahlfors, Lectures on quasiconformal mappings, Chap IV, where this is called $M$-condition, modern term is "quasisymmetric"), but for arbitrary homeomorphisms this is not true.
The answer to the other two questions is no. The procedure that you describe is called the Schiffer variation of conformal structure, and it indeed changes the conformal structure (of the resulting compact surface). See for example, S. Nag, Schiffer variation of complex structure and coordinates on Teichmuller spaces, Proc. Indian Acad. Sci., 94 (1985) 2-3, p. 111-122. The original source is M. Shiffer and D. Spencer, Functionals of finite Riemann surfaces. Princeton University Press, Princeton, N. J., 1954.