Is there such thing as a "smallest positive number that isn't zero"?

A simple proof by contradiction works here.

• Suppose that $a$ is the smallest positive real number.
• Next, divide it by $n$ (where $n>1$) to get $\displaystyle\frac a n$.
• This new number is smaller than $a$.

Your brother choose $n=2$, while you chose $n=10$.

So we can deny the existence of a smallest positive real number since

... there is a "smallest" number and yet there is a number smaller than it.

Same argument works with positive rational numbers.

You can put an infinite amount of zeroes in the decimal place before a number, (0.1, 0.01, 0.001 etc.) I am not entirely sure if our reasoning is correct though.

This claim is technically mistaken, which makes your brother's reasoning more correct than yours. You should recognize that the word "infinite" here is effectively just shorthand for "goes on forever", "doesn't have any end to it", and "always has another of the same digit coming up next". So it's a contradiction in terms to say that you can have an infinite number of zeroes and then some other digit afterward; this essential contradiction means that there's no real number like that, and thus, no smallest positive real number.

On the other hand: It would be correct to say that you can have an arbitrary number of zeroes before a 1, that is, indeed be able to find a positive decimal less than any other number someone proposed as "smallest".

There is no smallest positive real number. The argument of your brother is correct.

Your argument is also correct. As mentioned in comments your brother divides by $2$ while your argument amounts to dividing by $10$. Note though that it is better to say that there can be arbitrarily many $0$ rather than infinitely many. (One cannot have infinitely many $0$ and then the first $1$, or non-zero digit, in a decimal expansion. But there is no bound on the number of $0$ one can have before the first non-zero digit; also in total there can be infinitely many $0$, but not before the first non-zero one.)

Of course there is a smallest positive whole number/integer, it is $1$. The halving argument does not work here, as you cannot split $1$ into two positive whole numbers.

There are various ramifications of this and you might want to look into infinitesimals or ordered sets if you are curious about such things.

As for the smallest object in the world, this is a physics question, which has no definite answer as far as I know. But there are some theories where there is a smallest measureable length in some sense, see Planck length.