I believe you'll need to use universal instantiation on $(2)$. As a universally quantified premise, you can instantiate with $\,a\,$, as you did with premise $(1)$, (since the domain of all the quantifiers is the same) then use modus tollens when you need it, applied to the instantiated statement and using the assumption $\lnot R$.
Also, as currently numbered, I think you need to justify resolution for line $(7)$ by citing $(3)$ and $(6)$ (I'm assuming you're referring to resolution to $\,P(a)\,$ given the statements $\,\left[P(a) \lor Q(a)\right]\;$ and $\left[P(a) \lor \lnot Q(a)\right].$
The key here is that you want your steps (currently numbered) $4 - 8$ indented (to designate a "sub-proof", where $(4)$ is stated as an "assumption", then $5$ follows from $(4)$ and the (soon to be) instantiated step resulting from $(2)$.
Line $(9)$ should still be a statement about the constant $a$, not $x$ as written, citing sub-proof, $(4 - 8)$ (by assuming (4), we obtain (8)), i.e. $(4)\rightarrow (8)$ given subproof $4-8$.
From $9$ to $10$ then, you are correct, as is your justification of universal generalization.
$(1) + (4)$ do not imply $q\lor r$, so undo line $(5)$ We do have that $(3) + (4)$ imply $p$. If $p \lor q$, and $\lnot q$, then $p$. Perhaps this can be line $(5)$.
Suggestion for the next step $(6)$: from $(3),$ along with the premise $\lnot q \rightarrow (u \land s)$, it follows by modus ponens that $u \land s$.
(7) Now, extract $s$ from $u \land s$.
(8) Then introduce the conjunction: $p \land s$. $p$ is from your new (5th) step, and $s$ from step (7).
Now we can use the premise $p\land s \implies t$ and $p \land s$ from step (8) to conclude by modus ponens that $t$, as desired
Key point: If you haven't used a premise, think of how it might help get you from what you have to what you need to establish. Always keep the goal or target proposition in your mind.
Best Answer
It is just a premise. You cannot logically deduce that Socrates is a man - but it is something you assert as part of your proof i.e. a premise. Just like you assert all men are mortals.