Based on the question and follow-up comments, there appears to be uncertainty in your goals and student learning outcomes. If you clarify those goals, then it should be pretty apparent which string topics to cover.
Keep in mind that strings (or any other data structure, primitive data type, algorithm domain or just about anything else in computers) have no inherent value to anyone, especially students. They're a means to achieving other ends. The rule of thumb is to start with the goal, then teach only whatever is necessary for students to achieve that goal.
Perhaps your goals are to teach algorithms to improve critical thinking, prepare students for a CS degree in college and prepare them for job interviews (if job interviews still give Leetcode-style questions by the time they're entering the workforce). If so, determine their experience level and pick problems of an appropriate difficulty. Few high school students I know of are interested in or at a level where KMP or Sedgewick would be appropriate.
At a beginning level, students should be able to index, iterate, concat, split, join, slice, search, adjust case and be comfortable working with common string library functions in their language. Students in an algorithms-focused course can then reimplement these library functions from first principles. Then consider functions that build on top of these primitive operations such as palindrome and isogram detectors and command line projects like Mad Libs, Hangman (possibly no longer a politically viable name) and Wordle.
Keep in mind arrays and strings are pretty similar as sequences of characters, so you may wish to emphasize this and teach the two concepts more or less as one, especially if you're working with C.
Now, if your goal is student engagement with less of an interest in college prep, consider games, animations, creative coding and web apps. Here, teach strings on a purely pragmatic, as-needed basis rather than on an "every programmer should know" basis (paraphrasing Spolsky's quote which really refers to professional programmers in the early 2000s, not hobbyists or students). Algorithms are important, but if you're working in a high-level language and focused on high-level goals, they're a black box students can simply invoke.
Regex is probably too much, but if validation comes up in the process of making an app, I'll give students a simple regex pattern and explain the basics, like
*. I know professional programmers who don't touch regex, so it seems safe to file away. For the CS prep folks, a bit of exposure to the concept is fine in high school, but that's about it. I don't recall using regex at all in graduate CS even though I'm the sort of person who'd be looking for opportunities to use it.
Unicode is a dry, messy industry/"real world" problem that doesn't need to be discussed much in high school. It's better to hide the ugly, dumpster fire side of programming for a bit. Unicode is a topic that regularly brings full-grown adult programmers with years of experience to tears1. Students will have enough frustrations to keep them busy working within a well-scoped sandbox. Expand the bounds over time, try to avoid unnecessary contrivances and acknowledge when you are simplifying things, but try to keep the rails on the crib. Like dumping all of your problems and dark secrets on the first date, showing too much mess up front scares most people off.
I believe you're coming from a country and language where Unicode is a large part of everyday life. I live in the USA where we can pretty safely stuff Unicode under the table during high school and pretend it doesn't exist. Then when students enter the workforce, show them the mess, bait-and-switch style. By then, it's too late and they have to deal with uber-frustrating Unicode problems nobody else wants to touch so they can pay off their debt.
If you accept the terms and conditions and really want to go into Unicode, maybe an image-to-unicode converter or command-line interface that uses box-drawing characters would be exciting. You could maybe sneak some vegetables into the pizza if there are any pressing lessons you want students to learn. Emojis are another thing that seems fun and elegant until you try to use them in a real app and encounter the edge cases, but they can be OK for education2.
1. See this animation of Twitter giving an internal server error when tweeting a zero-width Unicode character.
2. As long as you're ready for them to inexplicably show up funny on Jane or Jimmy's computer and spend 3 hours trying to figure out why.