How do we know that there are not more numbers than there are names? I thought about this question for a longer time. There is a standard proof by contradiction that there is no smallest positive rational/real number $r$ by considering $r/2$.
Now what irritates me about that proof is that it relies on the assumption that such an $r$ or the algorithm for constructing such an $r$ could be given explicitly if it existed. Maybe it just can not but it still can exist. Maybe it can only be addressed by introducing a new way for the description of infinitely small numbers or, equivalently, the numbers that "lie between arbitrary large numbers and infinity".
If we do not have a way of talking about such a regime of numbers, then how can we comprehend a set like the natural numbers as it is one object with infinitely many numbered elements but not containing infinity? 
Normally, infinity is not included in the natural numbers because one would argue that if it was a number, then one could find the number just before infinity but that would not be possible. But what if we introduced a way of talking about such a number? A way of talking about a smooth transition from infinity to numbers that we can write down? Maybe the name or algorithm for the smallest positive decimal number would just take an infinite amount of time to be written down.
What if there are more numbers than there are names?
 A: It sounds like you have been confused by the colorful phrasing sometimes used to explain proof steps -- that is, things like, "for every number you can give me ..."
Speaking that way generally tends to make it easier for beginners to understand the structure of proofs, but it is also possible to take it too seriously, which you appear to be doing.
In particular, the actual content of a proof (say, a proof that there is no smallest positive number), does not really depend on anyone physically "giving" numbers to each other. As far as the proof goes, saying "every number you can give me" does not mean anything that's different from "every number" -- the "you can give me" is just a vivid way of reminding ourselves that the proof we're constructing is not allowed to rely on things that are only true for some numbers.
The assertion the proof depends on is simply that every positive number equals two times some other number, which is also positive but smaller than the first number. Note well that in this formulation, the claim does not speak about anyone doing anything -- it just states an eternal fact about numbers, not that we "can divide every number by 2", but that half of each number simply exists, has always done so and will always do so. And this general fact does not depend on whether the numbers have names or whether we can communicate each of the numbers exactly between each other.
Of course, if you desire to, you can start investigating what you can prove if you do insist on only speaking of things you can actually imagine doing -- that is, for example, only speak of numbers that can be communicated using finite description. This leads to constructive mathematics, which is an entirely respectable area of study. You just need to be aware that constructive mathematics is not what ordinary mainstream mathematics aspires to be: everyday proofs do allow things that cannot be justified constructively, and if the default assumption when you speak to mathematicians is that the proofs you speak about are according to the everyday non-constructive rules, however vividly they are phrased.
A: If a number can be defined, let its name be its (possibly long) definition. Then any number we can define has a name. (F.i., the name for $\sqrt{2}$ would be "the positive root of $x^2=2$".) 
A: Set theory has the concept of transfinite numbers (for example the first infinite ordinal $\omega$), so learning about that might answer some of your questions (I'm not sure anything would qualify as a "smooth transition from infinity to numbers that we can write down" -- you would need to describe more rigorously what you mean by that).
As for whether arbitrarily small positive rational numbers can be written down (in principle, given enough time and paper/ink), the answer is clearly yes. Suppose I have written down some rational number $p/q$ where $p$ and $q$ are in decimal digits; well, I just add a zero to the end of $q$ and I have a smaller number. So in that sense every positive rational number has a "name" (digits, followed by "/", followed by more digits) although that "name" may be very long.
As for whether every irrational number can be "defined", it appears to be more complicated (I think it may depend on what model of set theory you are presumed to be working in and what exactly you mean by "define"). This MO post discusses such issues.
Regarding some concept of number where $\delta > 0$ but $\delta / 2 = 0$: I doubt anything exactly like that exists. What is $\delta / 1.1$ in such a system? Or $\delta / 1.0000001$? However, as mentioned by another commenter, the hyper-real numbers are sort of like this.
