Can $\delta(t+\infty)$ be a legitimate signal?
It's possible, in mathematics, to complete real numbers with "infinite" values, with sound topological properties; for instance non-standard analysis or the Extended real number line (discussion at Math StackExchange).

However, in this context, for any standard real number $t$, the rule is to set $t\pm \infty = \pm \infty$ (see Arithmetic operations). So, using your notations, for every $t$, you would have [CAUTION ADVISED] $\delta(t+\infty)=\delta(\infty)$, to which I am not able to give a meaning, else than something being a constant evererywhere equal to $\delta(\infty)$. Of course, $\delta$ is a distribution, and should not be treated as a function, but defined as a functional operator, or via limits of functions.

However, so far, I have never seen  


a (serious and useful) use of the extended real system in signal processing,
a reference trying to define such a Dirac at infinity. 


For the latter, I fear (just intuition) that constructing this with test functions can be troublesome, since two different limits are required (location and amplitude). This is just an analogy, but $\delta(0)\times \delta(0)$ is, as far as I know, not defined.