Learn Hardcore Russian: У меня до сих пор сильное впечатление от той грозы над морем. - I still have a strong impression from that storm over the sea.

У меня literally means “by/at me” but in practice often corresponds to English “I have” or “for me” in a more abstract sense.
So У меня до сих пор сильное впечатление… is like saying “I still have a strong impression…” or “I am still under a strong impression…”.

Using я with иметь (e.g. Я имею сильное впечатление) is grammatically possible but sounds unnatural and overly formal in everyday speech. The у меня construction is the normal, idiomatic choice here.

Where is the verb in this sentence? Why is there no есть or имею?

In Russian, the present‑tense verb “to be” (быть) is usually omitted in simple statements.

Underlying structure: У меня (есть) сильное впечатление…
In normal speech/writing, есть is dropped: У меня сильное впечатление…

So the Russian sentence without an explicit verb corresponds to English “I still have a strong impression from that thunderstorm over the sea.”

You only must show the verb быть in:

the past: У меня было сильное впечатление…
the future: У меня будет сильное впечатление…
or when you really want to emphasize existence: У меня есть идея (“I do have an idea”).

What exactly does до сих пор mean, and can it go in a different place in the sentence?

До сих пор means “up to now, until now, still” and highlights that the state continues from the past into the present.

In this sentence:

У меня до сих пор сильное впечатление… = “I still have a strong impression…”

You can move до сих пор without changing the basic meaning:

До сих пор у меня сильное впечатление от той грозы над морем.
(A bit more emphasis on “even now / still”.)
У меня сильное впечатление от той грозы над морем до сих пор.
(Also possible; feels slightly heavier; many speakers will prefer it earlier.)

The most natural positions here are:

right after у меня: У меня до сих пор…
or at the very beginning: До сих пор у меня…

Why is it сильное впечатление, not some verb like “impressed”? How does this work in Russian?

Russian often uses nouns instead of adjectives or passive forms that English would use.

English: “I’m still very impressed by that storm…”
Russian: “I (have) a strong impression from that storm…” → У меня до сих пор сильное впечатление от той грозы…

Here:

впечатление = “impression” (a neuter noun)
сильное is the neuter singular adjective agreeing with впечатление

So, instead of saying “I am impressed,” Russian literally says:
“At/with me (there is) a strong impression…”

Why is it впечатление от and not впечатление о or something else?

The noun впечатление combines differently with от and о:

впечатление от чего? (Genitive)
= impression from some experience, event, performance, etc.
→ сильное впечатление от той грозы
“a strong impression from that storm”
впечатление о чём? (Prepositional)
= impression about something in a more general, “opinion” sense, often after you’ve seen/read/heard it.
→ моё впечатление о фильме
“my impression/opinion of the movie”

Here it’s about the emotional/experiential impact made by that particular storm, so от (“from”) with the Genitive is the natural choice:
впечатление от той грозы.

Why is it той грозы? What case is that, and why is it used?

Той грозы is Genitive singular, feminine, and it is required by the preposition от:

Preposition: от
- Genitive
Noun: гроза (storm, thunderstorm), feminine
- Nom.: эта гроза
- Gen.: этой грозы
With та (“that”):
- Nom.: та гроза
- Gen.: той грозы

So:

от той грозы = “from that storm”
(от forces той грозы into the Genitive case.)

Grammatically:

от → Genitive
та → той
гроза → грозы

What does над морем mean exactly, and why is морем in that form?

Над морем means “above/over the sea.”

Preposition над (“above, over”) takes the Instrumental case.
Noun море (sea), neuter:
- Nominative: море
- Instrumental: морем

So:

над морем = “over the sea”

This phrase describes the location of the storm:

та гроза над морем = “that thunderstorm over the sea”

Could we say на море instead of над морем? What’s the difference?

Yes, you can say на море, but the meaning changes:

над морем = “over the sea, above the sea”
Focuses on the sky above the sea as the place of the storm.
на море = literally “on the sea / at the sea”
Often used like “at the seaside” or “on the sea (surface)” in a broad sense.

Compare:

гром гремел над морем = “thunder rumbled over the sea”
(clearly in the sky)
мы отдыхали на море = “we vacationed at the seaside”

In your sentence, since we want the image of a storm in the sky above the sea, над морем is more precise and natural.

Is the word order fixed? Can I move parts of the sentence around?

Russian word order is flexible, though some versions sound more natural.

Your original:

У меня до сих пор сильное впечатление от той грозы над морем.
(Very natural.)

Other possible orders:

До сих пор у меня сильное впечатление от той грозы над морем.
– Emphasizes “still / even now” a bit more.
У меня сильное впечатление от той грозы над морем до сих пор.
– Possible, but до сих пор feels a bit tacked on at the end.
Сильное впечатление от той грозы над морем у меня до сих пор.
– Grammatically fine, with some emphasis on “a strong impression from that storm over the sea” as the topic.

While many orders are technically correct, the original and the first variant are the smoothest and most typical in neutral speech.

Is there a more common or more emotional way to say this idea in Russian?

Yes, a very common idiomatic variant is:

Я до сих пор под сильным впечатлением от той грозы над морем.

Differences:

Я… instead of У меня…
Now я is directly the subject.
под сильным впечатлением = “under a strong impression,” a very natural idiom.
The sense is slightly more emotional / expressive; it highlights the psychological state you are still in.

Both sentences are correct and natural:

У меня до сих пор сильное впечатление от той грозы над морем.
Я до сих пор под сильным впечатлением от той грозы над морем.

The second one may sound a bit more vivid in everyday speech.