Your text seems to have edited the original, which was written 125 years ago to sound like how certain upper-class people spoke in one time and place, to make the language more modern. But somewhere along the line, someone introduced an error. I think it was supposed to say,
However, it doesn't matter. Now that I look at the inscription inside, I see that this isn't yours."
You can’t omit the word “that” from this sentence, in American English. (Actually, as you’ve probably noticed and gotten frustrated, people often mumble or forget to say important words, and you just have to guess what they meant.)
When to use the simple present and when to use the present progressive can be tricky. Here, I think either way would work (“Now that I look at the inscription,” or “Now that I’m looking at the inscription”).
One time you’d have to use the present progressive is when you describe something that you’ve started but not finished doing. For example, let’s say that you’re playing the role of Algernon here. You get to the stage direction that says you’re supposed to open the case, but the prop in your hands is stuck, and you can’t get it to open. You might ad lib the line:
I think that is rather mean of you, Earnest—or maybe not. Now that I’m opening this thing, it’s harder than it looks. Your case might not be worth a reward after all.
You can’t say, “now that I open this thing,” because you aren’t able to open it yet. You’ve begun to open it, but you have not finished, so your action is in progress and you say “am opening.” However, you can say either, “Now that I try to open” it or “Now that I’m trying to open it.”
Or maybe the actor playing Earnest sees that the prop is stuck and asks, “Are you having trouble?” You might retort, “No, I open a case when I feel like it, and not a moment before.”
You can’t say either of those sentences the other way around. “*Do you have trouble?” is just not correct idiomatic English, and sounds wrong to native speakers, even though there’s no formal rule that it breaks. “I open a case when I feel like it,” has to be in the simple present because you’re talking about what you habitually or normally do, not what you happen to be doing at this moment.