Learn Hardcore Polish: Wieczorem często wspominam wakacje nad morzem. - In the evening I often remember the holidays by the sea.

Wieczorem is the instrumental form of wieczór and functions like an adverb of time.
Saying w wieczór is not natural in modern Polish in this meaning.

Can I change the word order and say Często wieczorem wspominam wakacje nad morzem?

Yes. Both are correct:

Wieczorem często wspominam…
Często wieczorem wspominam…

Word order in Polish is flexible. The difference is mostly about rhythm and emphasis:

Wieczorem często… slightly emphasizes the time (“in the evening”) first.
Często wieczorem… slightly emphasizes frequency (“often”) first.

Both will be understood the same in normal conversation.

Why is the verb wspominam used here and not something like pamiętam?

Both verbs relate to memory, but they’re different:

pamiętać – “to remember” (have something in your memory)
- Pamiętam nasze wakacje. – I remember our holidays.
wspominać – “to recall / to think back to / to reminisce about,” often with some emotion and as an active mental action.
- Wieczorem często wspominam wakacje… – In the evening I often think back to / reminisce about the holidays…

Here the idea is that you actively recall those holidays, not just that you have them in your memory.

What aspect is wspominam, and why that one?

Wspominam is present tense, imperfective aspect of wspominać.

wspominać (imperfective) – to be recalling / to recall habitually
wspomnieć (perfective) – to recall once, to mention once

In this sentence, you’re describing a repeated, habitual action (“I often do this”), so you must use imperfective:

Wieczorem często wspominam… – I often (habitually) think back…

A perfective form like wspomnę would mean “I will recall (once, at some point)” and doesn’t fit with często.

Why is wakacje used without any preposition? What case is it?

The verb wspominać takes a direct object in the accusative case:

wspominać kogo? co? – to recall whom? what?

Wakacje here is accusative plural, which happens to look the same as the nominative plural:

Nominative plural: wakacje – holidays
Accusative plural: wakacje – (I recall) the holidays

So wakacje is the object of wspominam, and no preposition is needed.

Why is it wakacje and not a singular like in English “holiday”?

In Polish, wakacje is a plural-only noun (pluralia tantum):

wakacje – school/university holidays, vacation (always grammatically plural)

You can’t normally say *wakacja in this meaning (except in some specific or joking usages). So even if in English you might say “my holiday,” in Polish you use the plural form:

Moje wakacje były wspaniałe. – My holiday(s) were great.

What does nad morzem literally mean, and what case is morzem?

Literally, nad morzem is “over/by the sea,” but in practice it means “at the seaside / at the sea.”

Grammar:

nad
- instrumental (for a static location) – “above / by / at (a body of water / river / etc.)”
morze – sea (nominative singular)
morzem – instrumental singular

So:

nad morzem – at/by the sea
Mieszkam nad morzem. – I live by the sea.

For movement towards the sea, you’d use nad + accusative:

Jadę nad morze. – I’m going to the seaside.

Could I say przy morzu or na morzu instead of nad morzem?

Not with the same meaning:

nad morzem – idiomatic for “at the seaside / by the sea (on land).” This is what you normally say about holidays.
przy morzu – literally “near/at the sea,” grammatically possible but sounds less idiomatic in this context.
na morzu – “on the sea,” usually means on a ship / out at sea, not on the beach.

For holidays on the coast, nad morzem is the standard, natural phrase.

Why can the pronoun ja be omitted before wspominam?

Polish is a pro-drop language: the personal pronoun is usually dropped because the verb ending already shows the person:

(Ja) wspominam – I recall
(Ty) wspominasz – you recall
(On/Ona) wspomina – he/she recalls

In neutral sentences you normally omit the pronoun:

Wieczorem często wspominam wakacje…

You would add ja only for contrast or emphasis:

To ty zapominasz, a ja wieczorem często wspominam wakacje.
It’s you who forgets, but I often think back to the holidays.

What is the function of często here, and where can I place it in the sentence?

Często means “often” and is an adverb of frequency modifying the verb wspominam.

Common, natural positions:

Wieczorem często wspominam wakacje nad morzem.
Często wieczorem wspominam wakacje nad morzem.
Wieczorem wspominam często wakacje nad morzem. (less typical, but possible)

What you generally don’t do is separate it far from the verb in a way that sounds unnatural, but overall Polish word order is flexible, and all of the above are understandable.