Is this correct arguing? Is this correct to conclude this way?

*

*Premise $\neg p$

*Premise $(\neg p \vee q) \to r$

*Therefore $\neg p \wedge (\neg p \vee q) \to r$

*Therefore $\neg p \to r$
 A: Yes, you are allowed to jump from step 3 to step 4, because the 2nd premise says that if either constraint $\neg p$ or $q$ is satisfied, then $r$ is implied.  In step 3, focusing on the left hand side, you notice that $\neg p$ is a premise.  Therefore, one of the constraints in premise 2 is satisfied, so the implication holds.
A: To deduce $\lnot p\to r$ you must be able to derive $r$ under the assumption of $\lnot p$.   So first assume that.
To derive $r$ under that assumption, using modus ponens with the premise of $(\lnot p\lor q)\to r$ as you were trying to do, you first need to be able to derive $\lnot p\lor q$.
Well, deriving $\lnot p\lor q$ under the assumption of $\lnot p$ is readily done by applying the rule of addition; also known as the rule of or-introduction, or disjunction introduction.
Now to apply modus ponens, you usually would not need to derive $(\lnot p\lor q)\land((\lnot p\lor q)\to r)$, but you obviously could, should your proof system require it.  But does it?
In any case, your proof was not too far off, and shows you have the right idea, but needs a lot of work.  So do try again before looking at the spoiler.

 ${1)~\lnot p\hspace{15.5ex}\textsf{by premise}\\2)~(\lnot p\lor q)\to r\hspace{5.25ex}\textsf{by premise.}\\\quad{3)~\lnot p\hspace{13.25ex}\textsf{by assumption}\\4)~\lnot p\lor q\hspace{9.5ex}\textsf{by addition }(3)\\4.5)~(\lnot p\lor q)\land((\lnot p\lor q)\to r)\hspace{4ex}\textsf{by conjunction $(4,2)$, should you need it.}\\5)~r\hspace{15ex}\textsf{by modus ponens }(4, 2)}\\6)~\lnot p\to r\hspace{10.75ex}\textsf{by deduction }(3{-}5)\\~\\\therefore \lnot p, (\lnot p\lor q)\to r\vdash \lnot p\to r}$

A: First, I would say you really need parentheses on line 3, i.e.



*Therefore $\neg p \wedge \color{red}((\neg p \vee q) \to r\color{red})$

Second, yes, all inferences are valid: line 3 follows from lines 1 and 2, and line 4 follows from line 3. So, in that sense it is 'correct', as user2661923 points out in their answer.
However, and third, ... few people would regard this as 'good reasoning', let alone a 'proper proof'. The whole idea of a proof is that you break it down to small steps that are intuitively obvious. In this case, few people would find the step from line 3 to 4 'intuitively obvious'. Indeed, you might as well have skipped line 3 altogether. This is why proof systems typically provide you with certain rules that reflect inferences that are 'intuitively obvious'. See GrahamKemp's answer for more details on that.
