I dont remember much so maybe you'll understand him better if you reread the story
Aside from not wanting to get outed, I feel like Jihwan's mindset to protect himself by thinking he's taken is just too deep for him to forget the act. Also he didn't know Sihyun is gay at the first place, he's thinking of safe answers throughout their relationship
Then as you already know, the lie keeps going because he cant stop and he seems to just let people to say stuff about him (by agreeing about the actor being his bf even tho he never thought of that lie until sihyun asked about it). Another example is how he didnt say anything when the female coworker said Jihwan always separate work and personal stuff, because thats what he wants others to believe, as long its a safe view for him, I guess. Thats why when he picked up the call and proved that it is personal matter, his coworker was shocked because its out of his character
I dont think it really hit him yet that he likes Sihyun tho, since he's really dense. But this is my opinion from what I remember, so I might be wrong
Oh no, I get where he is coming from. His character has been shown as someone who puts an act to be what other people perceive him to be or what he wants others to perceive him as, because he doesn't like showing weakness, he feels more in control and also, because he is scared of other people's opinion of him.
But it's just soooo frustrating ╥﹏╥ why can't he be honest...at least, halfway.
And also, he does like Sihyun. He had attraction towards him since the military, but in chapter 42 we see him questioning why he is angry at Sihyun for offering money after sex, even though that's the kind of man he is (someone who likes money and a good position, above stuff like feelings), and he seems to realize that he likes him (implicitly).
Mmm idk, Jihwan is too smart and above all, he is too conscious of himself. He might acknowledge that he likes Sihyun, but I see him trying to get the best of their current relationship until they both get tired of it and then forget those feelings (we all know that's not going to happen, but that might be his line of thought). Because Jihwan is too conscious of himself and he cares a lot about other people's opinion. It wouldn't be a problem for him if Sihyun was a woman (and if Jihwan himself liked women), but he isn't, and for Jihwan being in a relationship with Sihyun can be seen as more trouble than it's worth it. That's why I don't see him resolving the misunderstanding anytime soon ( ̄∇ ̄")

Why is Jihwan doing this? I mean, I get he is scared of being outed and etcetera but, why is he pushing himself deeper in this misunderstanding hole?
We know Sihyun had liked him since they were in the military, and was pissed off about Jihwan somehow playing with his feelings by saying their kiss was a mistake. He is just trying to be close to Jihwan however he can.
But what's Jihwan excuse? Likeee, why not say 'Oh, this ring is just for appearance since I like guys'? Why go down the rabbit hole of pretending to cheat? Even now, Jihwan knows he likes Sihyun but I don't see him (voluntarily) trying to resolve the misunderstanding (at least, not anytime soon).