Impact of modulation index in FM
Assuming an FM modulation of a sine wave of frequency $f_m$. Theoretically, the modulated signal is an infinite sum of cosines at the frequencies $f_c + nf_m$ ($n\in \mathbb{Z}$) which the amplitudes depend on the coefficients $J_n(\beta)$ (of course in addition to the amplitude of the carrier wave), where $\beta$ is the modulation index and $f_c$ is the carrier frequency. At this stage, the answer to the question Q.2. is yes. For the other items:


Q.1. The PSD of the modulated signal in $f_c$ is determined for $n=0$, then the PSD at that frequency depends on the coefficient $J_0(\beta)$. Knowing that $J_n(\beta)$ trends to zero as $\beta$ trends to infinity, so the PSD of the modulated signal in $f_c$ trends to zero as the modulation index trends to infinity.
Q. 3. Theoretically the bandwidth of an FM modulated signal is infinite. But when considering Carson bandwidth rule ($B_C$), the answer to your question is yes since this rule considers the modulation index. $B_C=2(1+\beta)W$ where $W$ is the baseband bandwidth of the modulating signal (assuming a sine wave, $W=f_m$).