Learn Hardcore Finnish: Toivottavasti onnistun sanakokeessa huomenna. - Hopefully I will succeed in the vocabulary test tomorrow.

Toivottavasti means hopefully (literally something like in a hoped-for way). It’s an adverb, so you can put it in front of a whole sentence: Toivottavasti onnistun... = Hopefully I succeed...
Toivon, että... means I hope that... and is a full clause with että: Toivon, että onnistun sanakokeessa huomenna.
Both are common; toivottavasti is shorter and a bit more “comment-like” on the whole sentence.

Why is there no word for I (minä) in the sentence?

Finnish usually doesn’t need subject pronouns because the verb ending shows the person.
onnistun already means I succeed (1st person singular), so minä is optional.
You might add minä for emphasis/contrast: Toivottavasti minä onnistun... = Hopefully I succeed (not someone else).

What is the dictionary form of onnistun, and what does the verb literally mean?

The dictionary form is onnistua = to succeed / to manage.
onnistun is the 1st person singular present tense form: (I) succeed / (I) manage.
So Toivottavasti onnistun... is present tense in form, but it often refers to the near future (here made explicit by huomenna).

Why does it say sanakokeessa (inessive “in”) instead of something like “on the test”?

sanakokeessa is sanakoe + -ssa (inessive case), literally in the vocabulary test.
With onnistua, Finnish commonly uses the inessive to mark the thing you succeed in/at:

Onnistuin kokeessa. = I did well on the test / succeeded in the test.
It’s not really about physical location; it’s a case pattern used with this verb.

What does sanakoe mean, and how is sanakokeessa formed?

sanakoe is a compound: sana (word) + koe (test/exam) → vocabulary test.
To make the inessive:

sanakoe → stem sanakoe-
- -ssa → sanakokeessa
  Notice an extra e appears before -ssa: that’s normal here (koe + -ssa → kokeessa).

Is sanakokeessa singular or plural? Could it mean “in vocabulary tests”?

It’s singular: in the vocabulary test.
Plural would be sanakokeissa = in (the) vocabulary tests.

What does huomenna do in this sentence, and where can it go?

huomenna means tomorrow and tells you when the succeeding is hoped to happen.
Finnish word order is flexible; you could also say:

Toivottavasti onnistun huomenna sanakokeessa.
Both are fine; the difference is mostly what you want to highlight.

Why is the verb in present tense (onnistun) even though it’s about tomorrow?

Finnish often uses the present tense for future situations when a future time word is present:

Huomenna onnistun. = Tomorrow I’ll succeed / I’m going to succeed.
So huomenna supplies the future meaning.

Could I also use pääsen läpi (“pass”) here, or does onnistun imply something else?

onnistun sanakokeessa generally means I do well / I succeed (good performance).
pääsen läpi means I pass (barely or just meet the requirement).
So:

Toivottavasti onnistun sanakokeessa. = hopefully I do well
Toivottavasti pääsen sanakokeesta läpi. = hopefully I pass the vocab test
(That second one also uses sanakokeesta “out of/from the test” in a common pattern.)

How is this sentence typically pronounced, especially the double consonants?

Key points:

Toivottavasti: the tt is long (held) — toi-vot-ta-vas-ti
onnistun: clear nn and st — on-nis-tun
sanakokeessa: stress on the first syllable SA-na-ko-kees-sa, and -eessa has a long vowel sound in kee
Finnish stress is almost always on the first syllable of each word.