Are there more rational numbers than integers?
Mathematicians have very precise definitions for terms like "infinite" and "same size". The single unambiguous correct answer to this question is that using the standard mathematical definitions, the rationals have the "same size" as the integers.
First, here are the definitions:
Define "$0$" = emptyset, "$1$" $= \{0\}$, "$2$" $= \{0,1\}$, "$3$" $= \{0,1,2\}$, etc. So, the number "n" is really a set with $n$ elements in it.
A set $A$ is called "finite" iff there is some $n$ and a function $f:A\to n$ which is bijective.
A set $A$ is called "infinite" iff it is not finite. (Note that this notion says nothing about "counting never stops" or anything like that.)
Two sets $A$ and $B$ are said to have the "same size" if there is a function $f:A\to B$ which is a bijection. Note that we do NOT require that ALL functions be bijections, just that there is SOME bijection.
Once one accepts these definitions, one can prove that the rationals and integers have the same size. One just needs to find a particular bijection between the two sets. If you don't like the one you mentioned in your post, may I suggest the Calkin-Wilf enumeration of the rationals?
Of course, these give bijections between the naturals (without $0$) and the rationals, but once you have a bijection like this, it's easy to construct a bijection from the integers to the rationals by composing with a bijection from the integers to the positive natural numbers.
You may not be very satisfied with this answer, but I'll try to explain anyway.
Countability. We're not really talking about whether you can "count all of the rationals", using some finite process. Obviously, if there is an infinite number of elements, you cannot count them in a finite amount of time using any reasonable process. The question is whether there is the same number of rationals as there are positive integers; this is what it means for a set to be "countable" --- for there to exist a one-to-one mapping from the positive integers to the set in question. You have described such a mapping, and therefore the rationals are "countable". (You may disagree with the terminology, but this does not affect whether the concept that it labels is coherent.)
Alternative mappings. You seem to be dissatisfied with the fact that, unlike the case of a finite set, you can define an injection from the natural numbers to the rationals which is not surjective --- that you can in fact define a more general relation in which each integer is related to infinitely many rationals, but no two integers are related to the same rational numbers. Well, two can play at that game: you can define a relation in which every rational number is related to infinitely many integers, and no two rationals are related to the same integers! Just define the relation that each positive rational a/b is related to all numbers which are divisible by 2a but not 2a+1, and by 3b but not 3b+1; or more generally respectively 2ka and 3kb for any positive integer k. (There are, as you say, sign issues, but these can be smoothed away.)
You might complain that the relation I've defined isn't "natural". Perhaps you have in mind the fact that the integers are a subset of the rationals --- a subgroup, in fact, taking both of them as additive groups --- and that the factor group ℚ/ℤ is infinite. Well, this is definitely interesting, and it's a natural sort of structure to be interested in. But it's more than what the issue of "mere cardinality" is trying to get at: set theory is interested in size regardless of structure, and so we don't restrict to maps which have one or another kind of "naturalness" about them. Of course, if you are interested in mappings which respect some sort of structure, you can build theories of size based on that: this is what is done in measure theory (with measure), linear algebra (with dimension), and indeed group theory (with index). So if you don't like cardinality as set theorists conceive it, you can look at more structured measures of size that you find more interesting!
Immediate predecessors. A somewhat unrelated (but still important) complaint that you make is this: "If it were possible to count to infinity, it would be possible to count one step less and stop at count infinity-1 which must be different to infinity." The question is: why would you necessarily be able to stop at 'infinity minus one'? This is true for finite collections, but it does not necessarily hold that anything which is true of finite collections is true also for infinite ones. (In fact, obviously, some things necessarily will fail.) --- This is important if you study ordinals, which mirrors the process of counting itself in some ways (labelling things as being "first", "second", "third", and so forth), because of the concept of a limit ordinal: the first "infinitieth" element of a well-ordering doesn't have any immediate predecessors! Again, you are free to say that these are concepts that you are not interested in exploring personally, but this does not mean that they are necessarily incoherent.
To summarize: the set theorists measure "the size of a set" using a simple definition which doesn't care about structure, and which may violate your intuitions if you like to take the structure of the integers (and the rational numbers) very seriously, and also want to preserve your intuitions about finite sets. There are two solutions to this: try to stretch your intuition to accomodate the ideas of the set theorists, or study a different branch of math which you find more interesting!
In mathematics a set is called infinite if it can be put into a 1-1 correspondence with a proper subset of it, and finite it is not infinite. (I know it seems crazy to have the concept of infinite as primitive and finite as a derivate, but it's simpler to do this, since otherwise you must assume that the integers exist before saying that a set is finite)
As for your remarks: - with your method (if you don't forget to throw out fractions like 4/6 which is equal to 2/3) you actually counted the rationals, since for each number you have a function which associates it to a natural number. It's true that you cannot count ALL rationals, or all integers; but you cannot either draw a whole straight line, can you? - with infinite sets you may build infinite mappings, but you just need a single 1-1 mapping to show that two sets are equal.