Does adding or/and dividing a random variable by a constant change its probability distribution?

Informally, if we have a random variable $X$, and $Y=aX+b$, where $a$ and $b$ are constants and $b\ne 0$, then $X$ and $Y$ are close relatives.

But the distributions of $X$ and $Y$ need not have the same type of name. As you pointed out, if $X$ has normal distribution, then so does $Y$. Similarly, if $X$ has uniform distribution, so does $Y$.

However, if $X$ has binomial distribution, then $aX+b$ only has binomial distribution if $a=1$ and $b=0$. Similar comments could be made about the hypergeometric, the Poisson, and many others.

The fact that if $X$ has binomial distribution, then (usually) $aX+b$ does not has no real mathematical significance. It just has to do with the kind of distributions we choose to call binomial. The very close relationship between $X$ and $aX+b$ remains, even if we do not happen to give their distributions the same name.