There are many non-isomorphic non-standard models of reals; any of them can be called hyperreals, although one specific model (the ultrafilter construction on $\mathbb{R}^\mathbb{N}$) is often called "the" hyperreals.
Models are generally taken to be sets. The surreal numbers are a proper class: they are "too big" to be considered a non-standard model of the reals in this sense.
But to some extent, we don't really have to insist on models being sets: with suitable set-theoretic axioms, I believe the surreal numbers are also a non-standard model of the reals. In fact, they would be the largest model.
If we pick one particular (set-sized) non-standard model -- e.g. "the" hyperreals -- then we cannot compare its elements to surreal numbers directly. First, we'd have to choose a way to embed the hyperreals into the surreals. There isn't a unique way to do this. In fact, there is a vast number of ways -- an entire proper class of embeddings! (I believe) we can choose to make any particular surreal a hyperreal number by choosing an appropriate embedding.
Numbers and Games
The $\{a,b\mid c,d,e\}$ notation used for surreal numbers is also used for representing certain games more generally. Basically, a "game" lets you put any sets of games as the left and right set. But a game is only a (surreal) number if all elements of those sets are numbers and no right element is less than or equal to any left element. For clarity, $*$ and $\uparrow$ are not (surreal) numbers, just games.
It turns out that numbers have nice properties: If $x=\{a\mid b\}$ is a number, then $a<x<b$ is true. However, that does not hold for games in general. So "greater than zero & less than star" is not a correct way of thinking of "$\{0\mid*\}$".
Definition of Inequality
To understand the meaning of inequalities and what incomparable would mean, we need a definition of inequality for games. There are a few equivalent definitions, but one that takes the least work to set up is given in Claus Tøndering's Surreal Numbers -- An Introduction. Paraphrased, Definition 2 says:
$x\le y$ if and only if $y$ is less than or equal to no member of $x$'s left set, and no member of $y$'s right set is less than or equal to $x$.
Now that we have this recursive definition of $x\le y$, we can define other (in)equality symbols:
- $x=y$ when $x\le y$ and $y\le x$ both hold.
- $x<y$ when $x\le y$ holds but $y\le x$ does not.
- $x\not\gtrless y$ ($x$ is "incomparable to" $y$) when neither of $x\le y$ and $y\le x$ hold.
You can see a notation-heavy use of this definition of $\le$ in this answer of mine explaining in detail how to check that $\{0\mid1\}$ is a number.
How can things be incomparable?
For example, consider the game (not a number) $s=\{1\mid-1\}$. If you check the definition of inequality above (or any equivalent one), you'll find that it's greater than $-2$ and less than $2$. But $s\le1$ and $1\le s$ are both false, so that $s$ is "incomparable to"/"confused with" $1$ (we might write $s\not\gtrless 1$). Similarly, $s$ is confused with $0$ (so "fuzzy") and confused with $-1$ as well. It is simply not true that "$s$ is somehow greater than $1$ and less than $-1$".
Your examples of $*=\{0\mid0\}$ and $\uparrow=\{0\mid*\}$ are similar. $*<1$ is true but $*\le0$ and $0\le*$ are not true (so $*\not\gtrless 0$). $0<\uparrow$ happens to be true, but $\uparrow\le*$ and $*\le\uparrow$ are not true (so $\uparrow\not\gtrless *$).
Best Answer
Overall, that PDF skips over some key clarifications about notation that may have led to some confusion. I would not recommend learning from it without external support like lectures or another more detailed text.
The context in the PDF is that this holds when $x$ is a number. If $x$ is some other game (like $\{0\mid*\}$), this equation can fail.
$∗+∗=\{∗\mid∗\}=0$ is true. But we need to take more care when evaluating something like $\emptyset+∗$. Since every game is an ordered pair of games, $\emptyset$ is not a game. However, there is a convention, used-but-not-explained in the addition section of that PDF for expressions that look like adding a set to a game. If $S$ is a set of games and $g$ is a game, $S+g$ is a shorthand for "the set of all games of the form $s+g$, for some $s$ in $S$". So $\emptyset+∗$ is a set, not a game like $0$. In particular, it's the set $\emptyset$.
As discussed above, $\emptyset+*$ is not $0$ -- it's not even a game. But you are right that "$g+*=0$ and $h+*=0$" would imply "$g=h$". In fact, you could add $*$ to both sides of $g+*=0$ to find $g+*+*=*$ so that $g+0=*$ and $g=*$. This idea works in general; negatives of games are unique (up to equality).
Just to emphasize one more time: $*$ is a game and $\emptyset$ is a set. They're different sorts of things.