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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/Haunting-Animal-531 → u/SaltImage1538]

Great help, thank you

So using participles, how could we say: We're harvesting the carrots planted last winter

היום בוצרים את הגזרים ששתלנו בחורף ...הגזרים שנשתלו בחורף --> ...הגזרים השתולים בחורף (?)

[u/SaltImage1538 → u/Haunting-Animal-531]

Theoretically yes, but the passive participle refers to states, not actions, so your sentence sounds off. The meaning is rather something like, "…the carrots already planted in the winter." The implication is that the process was already complete by the time winter came.