Yes you are right.

The impulse modulation description of ideal sampling operation introduces an amplitude scale of $1/T$ on the spectrum of the sampled signal, which is a periodic replica of the original bandlimited signal spectrum.

Note that as the sampling period $T$ gets shorter, the scale factor gets larger and the separation between repeating replicas gets larger too. So in the limit you will be left with a single copy of the original spectrum with infinite amplitude scale.

The mathematical definition of energy in the sampled signal is always infinite, regardless of the sampling period (except possibly at the limit of $T = \infty$).

This is an ideal operation. For the practical applications what you are interested in is the finite valued samples of the signal at the sampling instants; the sample values (envelope) will be the same irrespective of the sampling period.

You sound right.

The impulse modulation description of ideal sampling operation introduces an amplitude scale of $1/T$ on the spectrum of the sampled signal, which is a periodic replica of the original spectrum.

Note that as the sampling period $T$ gets shorter, the scale factor gets larger and the separation between repeating replicas gets wider too.

This is an ideal operation. In practice, what you are interested in is the finite valued samples of the signal at the sampling instants; the sample values (envelope) will be the same irrespective of the sampling period.