Sean's last comment inspired the following answer:
Let $100\epsilon < \inf |x - f(x)|$. Let $g(x) = \frac{1}{1 + 10\epsilon} f(x)$. Then by triangle inequality we have that $|x - g(x)| > \epsilon/2$.
Let $h: (1+10\epsilon)^{-1}B \to (1+10\epsilon)^{-1}B$ be the smooth map formed by
$$ h(x) = \eta* g(x) $$
where $\eta$ is a mollifier supported in $\epsilon B$. We have that $h(x)$ is smooth and has no fixed points etc.
Your statement of Theorem 4 is missing an assumption on $K$, such as being convex, or at least homeomorphic to such a set (convex, closed, bounded). Without such an assumption, rotation of a circle gives a counterexample. Also, I think that in Theorem 4 you want the normed space to be complete, i.e., a Banach space.
Theorem 3 is contained in Theorem 4, because on a compact set every continuous map is compact. Theorem 4 cannot be easily obtained from Theorem 3 (I think) because if we tried to simply replace $K$ with $\overline{f(K)}$ (which is compact), we can't apply Theorem 3 because $\overline{f(K)}$ is not known to be convex.
Both 3 and 4 were stated and proved by Schauder in his 1930 paper Der Fixpunktsatz in Funktionalraümen, which is in open access. Here is Theorem 3:
Satz I. Die stetige Funktionaloperation $F(x)$ bilde die konvexe, abgeschlossene und kompakte Menge $H$ auf sich selbst ab. Dann ist ein Fixpunkt $x_0$, vorhanden, d.h. es gilt $F(x_0)=x_0$.
And this is Theorem 4 (in slightly less general version: the image of $F$ is assumed compact instead of relatively compact; possibly because the latter concept wasn't in use).
Satz II. In einem "B"-Raume sei eine konvexe und abgeschlossene Menge $H$ gegeben. Die stetige Funktionaloperation $F(x)$ bilde $H$ auf sich selbst ab. Ferner sei die Menge $F(H)\subset H$ kompakt. Dann ist ein Fixpunkt vorhanden.
("B"-Raume is what is now called a Banach space.) So, it is correct to call both Theorem 3 and Theorem 4 "Schauder's fixed-point theorem".
And yes, Theorems 1 and 2 follow by specialization of Theorem 3 or 4 to finite dimensions.
Best Answer
There are contractible compacta without the fixed point property. In fact,
In a positive direction, here is a result which generalises John Palmieri's answer and follows directly from the Brouwer fixed point theorem. The idea hinges on the following observation.
To leverage this we have:
In fact every locally-finite CW complex of dimension $n$ embeds in $\mathbb{R}^{2n+1}$ as a neighbourhood retract (see Fritsch, Piccinini, Cellular Structures in Topology Th.1.5.15). Of course every compact CW complex is finite-dimensional, and has bounded image under the previous embedding. That a finite complex should be a retract of any disc in which it embeds is a consequence of Paul's answer here.
Now, in light of the previous two observations we use Brouwer's fixed point theorem to conclude the following.
Of course the full machinery is more powerful, and has application outside the realm of CW complexes.
The argument is as before. Show that the spaces in question embed in some disc as a retract and quote the Brouwer fixed point theorem.