I have the followig sets:
$$ A = \{ x \mid \varphi_x(x) = x^2 \} $$ $$ B = \{ x \mid \varphi_x(x) = 10 \} $$ $$ C = \{ x \mid x \in E_x\} $$
These are not index sets, therefore Rice's Theorem and many-to-one reductions cannot be used to prove their undecidability. What kind of proof could be used?
Rice's theorem can't be used, but many-one reductions absolutely can be used.
For example, looking at $A$, consider the following. For each $n$, let $M_n$ be a machine which on input $x$ runs $\varphi_n(n)$; if $\varphi_n(n)$ ever halts then $M_n$ halts and outputs $x^2$, and if $\varphi_n(n)$ never halts then $M_n$ never halts either. The function $\mu_n$ calculated by $M_n$ clearly is $x\mapsto x^2$ iff $n$ is in the halting problem, and there is an obvious computable map $f$ such that $\varphi_{f(n)}\simeq \mu_n$, so we get a many-one reduction of the halting problem to $A$.
There are situations where we cannot use many-one reductions; often these are addressed by considering coarser reduction notions (e.g truth-table or Turing), while another common technique is diagonalization. But each of the problems in your question can be solved with many-one reductions alone, despite not being index sets.