Can someone help me from where in those rules I can deduce what is printed below, i.e. that if $\Gamma\vdash b:\mathsf{Glue}[\phi\mapsto(T,f)]A$, then $\Gamma,\phi\vdash b:T$? All the gluing types are introduced right before the paragraph (see PDF), the only preceding definition used here is $\Gamma\vdash a:A[\phi\mapsto u]$ (like in the top right rule for $\mathsf{unglue}$), but I as far as I understand, $\mathsf{Glue}[\phi\mapsto(T,f)]A$ is purely syntactical, i.e. it's just the type $\mathsf{Glue}$ with parameters $\phi, T, f, A$.
Only thing I can imagine is that is has something to do with the rule $$\Gamma\vdash\mathsf{Glue}[1_\mathbb{F}\mapsto(T,f)]A=T$$ if the $\phi$ could be baked into $\Gamma$, but as far as I understand, the face lattice is a seperate object, so one couldn't e.g. define $\Gamma':=\Gamma,\phi$.
Also, are gluing types meant to be inductive, e.g. when $b:\mathsf{Glue}[\phi\mapsto(T,f)]A$, do I know that $b$ is constructed via $\mathsf{glue}$ because this is the only constructor in $\mathsf{Glue}$? (I assume not, just asking to be sure...)
You are correct about which rule entails it.
First, contexts $Γ$ don't just list variables, but 'face assumptions.' So $Γ,\varphi$ itself is a valid context which you could call $Γ'$ or something, and it means 'whatever $Γ$ means, and also $\varphi$ is true.'
Next, under the assumption $\varphi$, $\varphi = 1_F$. So, in the context $Γ,\varphi$, the system $[φ \mapsto (T,f)]$ is equivalent to $[1_F \mapsto (T, f)]$. And this allows the rule you mentioned to trigger.
Sometimes multiple contexts are used, one for 'normal' variable assumptions, and another for face/"cofibration" assumptions. But I think in the paper you're reading, they're just all merged into one context.
As for your second question, I believe $\mathsf{Glue}$ is arguably coinductively defined. You can find a discussion of that sort of thing in section 3.2 of this paper. I think it doesn't make much actual difference for non-recursive types like $\mathsf{Glue}$. But in some sense, the intuition is not that every element is necessarily 'made of' a $\mathsf{glue}$ constructor, but every element can be presented that way, because of an eta rule where you $\mathsf{glue}$ together what you get from $\mathsf{unglue}$ on any given value. But the 'projections' ($\mathsf{unglue}$ and the system) are supposed to be primary.