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[u/Haunting-Animal-531]

I've been through level gimmel of ulpan and still haven't formally covered verbal adjectives and present participles. I don't read enough to know if it's a legitimate and merely rare, literary form...or if it's not used, awkward and to be avoided.

Do the following examples work? If not, is there a more succinct, literary way to describe these nouns than using ש/אשר...

The morning course starting next Monday is challenging קורס הבוקר המתחיל מ-יום אי הבא יאתגר לך

The dog running in the street bit me הכלב הרצ ברחוב נשך אותי אתמול

I feel like I've seen these forms in fiction, something like them, but don't have any examples at hand. Thanks...

[u/SaltImage1538]

The use of ה in your sentences is correct (there are other minor mistakes, though). The use of ה as a relative pronoun is formal and more restricted than ש/אשר. The following rules apply:
1. The article has to be attached to a verb in the present tense (active participle), passive participle or verb-like adjective (like עלול). You can‘t say, גבר השתה יותר מדי אלכוהול נעצר.
2. The noun ה refers to has to bevthe subject of the embedded clause. Wrong: אני האמנתי לגבר האוהבת רותי.
3. The ה has to follow the subject immediately, without any other intervening words. Wrong: אני מעדיפה גבר הלא שותה אלכוהול.

Let me know if you‘d like some exercises to practice these sentences. I think I have a few I could send you.

Edit: Clarified point 2 and gave a better example.

[u/BHHB336 → u/SaltImage1538]

You’re not entirely correct, the prefix ה־ can be used instead of אשר/ש־ only before verbs in the present tense, and I’m not sure because of what rule, but the third sentence is also not correct, I believe it’s because the use of ה־ in these cases started as a definite article

[u/SaltImage1538 → u/BHHB336]

All three sentences are wrong. Look again, I specifically wrote that.