(1). There are endomorphisms $T$ with $\ker(T)=\{0\}$ which are not surjective.
(2). Not in every case a linear form $\phi$ is representable by a vector $v$ in presence of a scalar product, i.e., there doesn't exist a vector $v$ that $\phi(.)=\langle v,.\rangle$.
(3). Not all linear mappings are continuos.
(4). You can equip a vector space with two different norms such that the unit ball in respect to the first norm in unbounded in respect to the second.
It's just a brand new world.
Let me answer your last two questions first:
In the example of $Q[t]$, what I would call the "standard basis" is $\{1, t, t^2, t^3, \dots \}$. This is a basis because every polynomial in $Q[t]$ can be uniquely expressed as a linear combination of those elements. But there are many other (in fact, infinitely many) bases for $Q[t]$; for example, $\{1, 1+ t, 1 + t +t^2, 1 + t + t^2 + t^3, \dots \}$ is also a basis because, again, every polynomial can be uniquely expressed as a linear combination of those elements. (As an exercise, can you express, say, $t^2 +3t -1$ as a linear combination of the elements in that basis?) Remember, vector spaces generally have many different bases.
Now on to your first question. As other answers have said, every vector space $V$ has a basis, which is a set of vectors $B$ such that every vector in $V$ can be uniquely written as a linear combination of vectors in $B$. (Note: by definition, linear combination means a finite sum.)
Unfortunately, for a lot of commonly occurring infinite dimensional spaces (e.g., function spaces such as the one you mentioned), there's no good way of writing down an explicit basis.
One last note: when one studies Hilbert spaces (which are "complete inner product spaces"), one uses the term "basis" (or "orthonormal basis") to mean something subtly different from our use of "basis" above. In that context, you are allowed to take infinite sums of "basis" elements. This confused me when I first studied Hilbert spaces.
Best Answer
Infinite-dimensional vector spaces are general enough that there is not a whole lot of interesting theory about them. To get anywhere you need to make some restrictions to the subject.
Probably where you want to go is functional analysis - the study of (usually infinite-dimensional) vector spaces with topological structure. As usual, Lang has an introductory book - Real and functional analysis - that could be helpful.