Is it the case that for every $n$, $m$, there is an integer expressible as a sum of $m$ distinct positive integer squares in $n$ distinct ways?

Question

I am not a mathematician, but I learn from various sources that we know that, for every n, there is an integer that can be expressed as a sum of two positive integer squares in n distinct ways, and similarly for cubes and fourth powers. I also learn that we do not know whether this goes for fifth powers (and every $k>5$?).

My question concerns a generalization of the first proposition: is it the case that,

1. for every $n$, and every $m$, there is an integer that can be expressed as a sum of $m$ distinct positive integer squares in $n$ distinct ways? And

2. the same for squares? And

3. the same for fourth powers?

I have added ‘distinct’ to prevent its being trivial by adding the cube (or square or fourth power) of zero. Since I am not a mathematician, answers will have to be comprehensible to a non-mathematician (but the answers could just be instances of 'yes' or 'no'). Thanks very much!