Learn Hardcore Russian: Мой приятель любит рассказывать шутки в трамвае. - My buddy likes to tell jokes on the tram.

друг = close friend, good friend, often with emotional closeness.
приятель = buddy, pal, mate, acquaintance-friend somewhere between знакомый (acquaintance) and друг (close friend).

Using мой приятель sounds a bit lighter / more casual than мой друг.

Why is it мой приятель and not моя приятель?

Because приятель is grammatically masculine in Russian.

Masculine singular мой приятель (my male friend/buddy)
Feminine singular would be моя приятельница (my female friend/buddy)

The possessive pronoun agrees in gender, number and case with the noun, so masculine приятель takes masculine мой.

How does любит рассказывать work grammatically? Why not just one verb?

In Russian, the structure [verb of liking] + infinitive is very common:

любить (to like / to love) + infinitive = to like doing something.

Here:

любит = he (my friend) likes / loves
рассказывать = to tell, to narrate (infinitive)

So любит рассказывать literally is likes to tell.
You can use the same pattern with other verbs:

Я люблю читать. – I like to read.
Она любит петь. – She likes to sing.

Why is рассказывать used here instead of говорить or шутить?

Subtle difference in meaning:

рассказывать шутки = to tell jokes (to narrate them, like little stories).
говорить is more general: to speak, to say. говорить шутки is not natural.
шутить = to joke, to be joking (not necessarily telling a full joke story).

Examples:

Он любит рассказывать шутки. – He likes telling jokes (full jokes).
Он любит шутить. – He likes joking (making jokes in general).

In your sentence, the idea is about telling actual jokes, so рассказывать шутки is the most natural.

Why is шутки in the plural here and not шутку?

Plural шутки emphasizes that he generally tells jokes (in general, as a habit), not just one specific joke.

любить рассказывать шутки – to like telling jokes (as an activity).
любить рассказывать одну шутку – to like telling one (same) joke (would sound odd here).

So the plural makes it a general, habitual action.
Grammatically, шутки is the accusative plural of шутка (feminine noun).

What case is шутки in, and how is that form built?

Шутки here is in the accusative plural.

Base noun: шутка (feminine, -а ending)

Declension (most relevant forms):

Nominative singular: шутк-а
Accusative singular: шутк-у
Nominative plural: шутк-и
Accusative plural: шутк-и

For inanimate feminine nouns like шутка, the accusative plural = nominative plural.
So рассказывать (что?) шутки – to tell what? jokes – accusative plural.

Why is it в трамвае and not на трамвае? What is the difference?

Both are correct but mean different things:

в трамвае = inside the tram (location, where he is physically).
на трамвае = by tram, by tramway (means of transport).

So:

Он рассказывает шутки в трамвае. – He tells jokes while he is in the tram.
Он ездит на трамвае. – He travels by tram.

Your sentence focuses on the place where he tells jokes (inside the tram), so в трамвае is used.

What case is трамвае and why does it end with -е?

Трамвае is in the prepositional case singular, governed by в (in, at) when talking about location.

Base noun: трамвай (masculine, ending in -й)

Main forms:

Nominative singular: трамва́й
Prepositional singular (в/на чём?): трамва́е

When a masculine noun ends in -й, the stem consonant + й typically changes to е in the prepositional:

музе́й → в музе́е
трамва́й → в трамва́е

Can I change the word order, for example: В трамвае мой приятель любит рассказывать шутки? Is it still correct?

Yes, that word order is correct and natural.

Russian word order is flexible. All of these are grammatically fine:

Мой приятель любит рассказывать шутки в трамвае.
В трамвае мой приятель любит рассказывать шутки.
Мой приятель в трамвае любит рассказывать шутки.

The differences are mostly about emphasis:

Starting with В трамвае emphasizes the location: In the tram, my friend likes telling jokes.
Original order is neutral: it just states what your friend likes to do and where.

Is there any difference in meaning between любит рассказывать шутки and something like часто рассказывает шутки?

Yes:

любит рассказывать шутки – focuses on his attitude: he enjoys it, he likes doing that.
часто рассказывает шутки – focuses on frequency: he tells jokes often, but we say nothing directly about whether he likes it.

You can also combine both:

Мой приятель любит рассказывать шутки и делает это очень часто.
My friend likes telling jokes and does it very often.

Could I leave out мой and just say Приятель любит рассказывать шутки в трамвае?

You could, but it would sound odd or incomplete in most contexts.

Мой приятель... clearly means my friend.
Bare приятель without a determiner usually needs some context (e.g., already mentioned earlier) or sounds like a very general type: a friend as a category, which is unusual.

Normally, when introducing your specific friend, you keep мой:

Мой приятель любит рассказывать шутки в трамвае. – natural.
Приятель любит рассказывать шутки в трамвае. – grammatically OK, but you would expect context like: Этот приятель любит... / Один приятель любит...

How do you correctly stress and pronounce the words in this sentence?

Stress (accented syllables in caps):

МОй приЯтель лю́бит расСКА́зывать Шу́тки в трамВА́е.

More precisely:

Мой – [moy], one syllable, stressed.
приятель – pri-YA-telʹ (при-я́-тель), stress on я.
любит – LYU-bit (лю́-бит), stress on лю́.
рассказывать – ra-SKA-zy-vatʹ (рас-ска́-зы-вать), stress on ска́.
шутки – SHUT-ki (шу́т-ки), stress on шу́т.
в трамвае – v tram-VA-e (в трам-ва́-е), stress on ва́.