(Parts of this will overlap what others have posted.)
You know how in English, if we’re doing something, we use the verb “do”? And if we are saying we are able to do something, we use the word “can”? (Related but off topic: And if we simply say we intend to do something, we use the word “will”?)
In Japanese, the generic verb for “do” is する. But when expressing that one “can do” (is able to do), the verb is できる. You may see a Japanese to English dictionary list a whole bunch of English translations for できる, but a lot of it will be context-dependent. At its core, できる is “can do”. (Or in the case of past tense, “was able to do”. English can be a bit wordy like that.)
Following this character’s できる is the ちゃう, which can also take on different meaning based on context. But the core meaning of ちゃう (which comes from て＋しまう) is that an action (of the verb it’s attached to) has completed.
When you have 「verbてしまう」 (or 「verbちゃう」), as I understand it, it typically means something happened or took place without you intending for it to. In some contexts, it can be regrettable. But in this case it’s a happy event, as the girl was able to have gotten a boyfriend.
Here’s the Cure Dolly video @wanikani_94032 mentioned, which I think is a great video on the topic:
Using Cure Dolly’s “done” translation, you can say the girl done went and got herself a boyfriend.
Based on my reading experience, I would never figure it as the で particle. I can only see it as できる. In other words, you’ll get used to it over time. It just takes a lot of immersion, seeing many different sentences in many different contexts over time.
And just for fun, できる can be written in kanji as 出来る =D
I think やっぱ is short for やっぱり, as they’re used the same.
なし is similar to ない, but whereas ない is an adjective, なし works like a noun. This allows it to be followed by a particle such as に or で.
Edit: Extracurricular reading on the origin of なし.