Can operations such as Modus Ponens be applied to variables used in an OR statement? Essentially, I want to know if I can something like the following is valid:

*

*$(C \rightarrow D) \lor F$

*$C$

*$(D) \lor F$   (By Modus Ponens on 1,2)

 A: Yes , $(C \rightarrow D )\lor  F \iff (\lnot C \lor D) \lor F \iff \lnot C \lor ( D \lor F)  \iff C \rightarrow ( D \lor F)  $
NOTE= $ (\lnot C \lor D) \lor F \iff \lnot C \lor ( D \lor F)  $ by the property of commutativity of disjunction
So by $1,2$ Modus Ponens (MP) ,we obtain $( D \lor F)  $
A: Careful!!!!!
You are effectively asking whether or not you can generalize inference rules so that they can be applied to component statements.
Well, for something like Modus Ponens, it turns out that yes, you can generalize it to the the following rule, which we might call 'Embedded Modus Ponens'): If you know the truth of some statement $\varphi$, and you have some other statement that contains $\varphi \to \psi$ as a component statement, then you can infer the statement that has $\varphi \to \psi$ replaced with just $\psi$
However, for other inference rules, this may not work. Consider the rule of Simplification (often referred to as $\land$ Elimination):

*

*$C \land D$

*$D$ (by Simplification on 1)

OK, no problem there. But now consider the following:

*

*$(C \land D) \to Q$

*$(D) \to Q$ (by 'Embedded Simplification' on 1)

Well, this argument is invalid! (I am sure you can verify this yourself)
Now, you could go through all the inference rules and document which ones can be generalized so that they also work on component statements. However, because of exceptions like the latter, it is much easier and safer to simply rule out the application of inference rules on component statements altogether, which is what most formal systems of logic do. Indeed, as such, your inference from 1 and 2 to 3 simply wouldn't be seen as an application of Modus Ponens, even though it is valid and 'feels' like Modus Ponens.
Long story short:
Is applying rules to component statements semantically valid (as in: does result logically follow)? Answer: it depends on the rule
Is applying rules to component statements syntactically valid (as in: does it conform to the rules as defined by the formal inference system)?  Answer: for most systems, no (and to make this concrete: if you do this on your HW, you will probably lose some points!)
A: If you learned Fitch style deduction system with inference rules, certainly you can correctly use Modus Ponens in a subproof which starts with its first case $(C \rightarrow D)$ after disjunction elimination for $(C \rightarrow D) \lor F$. After you arrive at $D$ at the end of this subproof, formally you may need disjunction introduction to further arrive at your goal $D \lor F$. An you can do same disjunction introduction to arrive at the same goal for the other case starts with $F$. Then it's strictly and formally done.
