Are you sure about number 1?
In another of your questions, Injective Equivalence, you have already shown that for conditions (2) and (3), equality holds just in the case where $f$ is injective. So, for examples where the inclusion is strict, look for functions which are not injective.
(The simplest non-injective function is the function $f:\{0,1\}\to \{a\}$ given by $f(0)=f(1)=a$. This is useful to know when constructing counterexamples.)
Number 4 is similar.
As an aside, your terminology here is a bit off:
I do know that to be a proper class one class is strictly contained within a larger class and excludes some of its members.
You mean a proper subset (or occasionally a proper subclass). A proper class is something altogether different.
Another way to define a proper subset is to say that $A$ is a proper subset of $B$ if $A\subseteq B$ and $A\neq B$. For this reason, some people use the notation $A\subsetneq B$ (to avoid the ambiguity over "$\subset$" - see here: $\subset$ vs $\subseteq$ when *not* referring to strict inclusion).
See :
Définition 1 (L'inclusion). La relation désignée par $(\forall z)((z \in x) \implies (z \in y))$ dans laquelle ne figurent que les lettres $x$ et $y$, se note de l'une quelconque des manières suivantes : $x \subset y, y \supset x$,
« $x$ est contenu dans $y$ », « $y$ contient $x$ », « $x$ est un sous-ensemble de $y$ ».
See English translation :
Regarding origins :
According to Florian Cajori (A History of Mathematical Notations (1928), vol. 2, page 294), the symbols for "is included in" (untergeordnet) and for "includes" (übergeordnet) were introduced by Ernst Schröder : Vorlesungen über die Algebra der Logik. vol. 1 (1890).
In addition, Schröder uses $=$ superposed to $\subset$ for untergeordnet oder gleich, i.e. $\subseteq$; see Vorlesungen.
Giuseppe Peano, in Arithmetices Principia Novo Methodo Exposita (1889), page xi, uses an "inverted C" for inclusion :
Signum $\text {"inverted C"}$ significat continetur. Ita $a \ \text {"inverted C"} \ b$ significat classis $a$ continetur in classis $b$ [i.e. : $\forall x(x \in a \to x \in b)$].
Note. It is worth noticing that in Peano there is the distinction between the relation : "to be an element of" ($\in$) and the relaion : "to be included into" ($\subset$).
This distinction is not present in Schröder.
According to Bernard Linsky, Russell’s Notes on Frege’s Grundgesetze Der Arithmetik from §53, in Russell, 26 (2006), page 127–66 :
Gregory Moore reports that Russell used $\supset$ for class inclusion as well as implication until March or April 1902, when he started to use $\subset$ for class inclusion.
Best Answer
Different people use different conventions. Some people use $\subset$ for proper subsets and $\subseteq$ for possible equality. Some people use $\subset$ for any subset and $\subsetneq$ for proper subsets. Some people use $\subset$ for everything, but explicitly say "strictly proper" in words when they feel it matters. I do not believe that there is a consensus for the meaning of $\subset$. My own personal advice is to use $\subseteq$ and $\subsetneq$ when you care to be precise, and $\subset$ when you are feeling lazy.