Is your claim 1 true?
Let's look at an agent who's Utility is quasi linear in money, $m$, and goods $c$.
The utility function is given by
$$U(m,c) = m + f(c)$$ for some concave function $f$.
The maximization problem, given wealth $w$, prices of money and consumption $p, p_c$ is then
$$\max_{m,c} U(m,c) \text{ s.t. } pm + p_c c = w\\
= \max_c f(c) + \frac{w}{p} - \frac{p_c}{p}c\\
\Rightarrow f'(c) = \frac{p_c}{p}
$$
So the optimal amount for consumption $c$ is independent of the wealth level $w$, if the interior solution qualifies as the global solution - which it does for sufficiently high wealth level. I'm not from a micro background but I suppose this qualifies as no income effect for the nonlinear good.
The intuition is, that, beyond a threshold of income, you have your optimal consumption locus $c^*$ which is identified by the previous FOC. Having a higher income will make you want to save that money, not put it into consumption.
What about the inverse?
Now that we got the intuition, we can try to work it backwards. What do we need? That
$$\exists c^* : U(m+\epsilon, c^*) > U(m, c^* + \frac{p}{p_c}\epsilon) \,\forall \epsilon > 0$$
One can show (do it!) that if $U$ was concave in $m$, eventually the marginal value of increasing $m$ (for a large enough $\epsilon$) was smaller than investing it into $c$.
On the other hand, if $U$ was additively separable in $(m,c)$ and increasing but convex in $m$, we would - after a threshold - always invest all into $m$ and set $c=0$. Also here, after that threshold, the optimal locus for $c$ is trivially independent of the wealth level: no income effect.
Another example would be decreasing utility in $c$, which would make us set it independently of the wealth level to its global minimum (often 0).
You see, unless you give us a class of utility functions, there can be many valid (but economically stupid) cases of utility functions where the inverse does not hold.
