The first one is essentially correct, though you should use $\mathbf{0}$ (boldface) to distinguish it from the scalar $0$, just like the original does.
It is more common to put quantifiers first and the clause to which they apply following them, in parenthesis. So you would write the second one as $\forall u,v\in W (u+v=v+u).$ That said, this does not correspond to the second statement above. Instead, what it says is that addition is commutative for elements of $W$ (it says: if $u$ and $v$ are any vectors in $W$, then the result of adding $u$ to $v$ is the same as the result of adding $v$ to $u$, which is not "any linear combination of $u$ and $v$ is an element of $W$).
Instead, you want to say something like (here $K$ is the field of scalars) $\forall \alpha,\beta\in K\Biggl(\forall \mathbf{u},\mathbf{v} \Bigl( \mathbf{u},\mathbf{v}\in W\longrightarrow \alpha \mathbf{u}+\beta\mathbf{v}\in W\Bigr)\Biggr).$ This says: for any two scalars $\alpha$ and $\beta$, and any two vectors $\mathbf{u}$ and $\mathbf{v}$, if $\mathbf{u}$ and $\mathbf{v}$ are both in $W$, then (their linear combination) $\alpha\mathbf{u}+\beta\mathbf{v}$ is also in $W$.
For the third, it's very similar to the second (in fact, I'm surprised it is included, since it is a special case of the second one, taking $\mathbf{v}=\mathbf{0}$): $\forall c\in K\Bigl(\forall \mathbf{u}\bigl( \mathbf{u}\in W\longrightarrow c\mathbf{u}\in W\bigr)\Bigr).$ This says: for every $c\in K$, for every vector $\mathbf{u}$, if $\mathbf{u}$ is in $W$, then $c\mathbf{u}$ is in $W$.
The above are a bit of "pidgin logical notation" rather than completely formal, because of the use of things like $\forall \alpha,\beta\in K$; but it is a common (ab)use of the notation.
If you really wanted to be more formal, the quantifiers should only have one variable each, and the conditions would be put in the statement. So the third statement could be $\forall c\forall\mathbf{u}\Bigl( \bigl((c\in K)\land (\mathbf{u}\in W)\bigr) \longrightarrow (c\mathbf{u}\in W)\Bigr),$ (you can put the universal quantifiers in the other order) and the second could be $\forall\mathbf{u}\forall\mathbf{v}\forall\alpha\forall\beta\Bigl(\bigl( (\mathbf{u}\in W)\land(\mathbf{v}\in W)\land (\alpha\in K)\land (\beta\in K)\bigr)\longrightarrow (\alpha\mathbf{u}+\beta\mathbf{v}\in W)\Bigr).$