После прогулки мой друг предложил мне чай.

Because после always requires the genitive case.

• Dictionary form: прогулка = walk
• After после: после прогулки = after the walk / after a walk

So прогулки here is the genitive singular form of прогулка.

It can mean either, depending on context.

Russian often does not use articles like a and the, so после прогулки could be understood as:

• after a walk
• after the walk

The wider situation tells you which is more natural.

Мой means my, and it agrees with друг in gender, number, and case.

• мой = my
• друг = friend

Together: мой друг = my friend

Russian often allows possessive words to be omitted if they are obvious from context, but here мой is included because the speaker explicitly says my friend.

Предложил is:

• past tense
• masculine singular
• from the infinitive предложить = to offer / to propose

It is masculine because друг is masculine.

Compare:

• он предложил = he offered
• она предложила = she offered

So мой друг предложил means my friend offered.

Because мне is the dative case, which is used for the person something is given, shown, offered, or said to.

• я = I
• меня = me (often accusative or genitive)
• мне = to me

In предложил мне чай, the meaning is offered tea to me, so the indirect object is in the dative:

• мне = to me

Here чай is the direct object of предложил, so it is in the accusative case.

But for inanimate masculine nouns, the accusative singular looks the same as the nominative singular.

So:

• nominative: чай
• accusative: чай

That is why the form does not visibly change.

In this sentence, the most natural meaning is offered me tea.

The verb предложить can mean:

• to offer
• to propose
• to suggest

When the object is something like чай, кофе, помощь, it often means offer.

So предложил мне чай is best understood as offered me some tea or offered me tea.

Russian has no articles, and it often leaves quantity unspecified unless it matters.

So чай here can mean:

• tea
• some tea
• sometimes even a tea in a looser English translation, though some tea is more natural

English needs a determiner more often than Russian does, but Russian can leave it unstated.

No, Russian word order is fairly flexible.

После прогулки мой друг предложил мне чай is a normal, neutral order:

• time expression first
• subject next
• verb
• indirect object
• direct object

But other orders are possible, for example:

• Мой друг после прогулки предложил мне чай.
• Мне мой друг после прогулки предложил чай.
• Чай мне предложил мой друг после прогулки.

These versions can sound more marked or put emphasis on different parts. The original sentence sounds natural and neutral.

Because после прогулки is just a normal prepositional phrase at the beginning of the sentence, not a separate clause.

Russian does not normally put a comma after a simple introductory time phrase like this.

So this is correct:

• После прогулки мой друг предложил мне чай.

Предложил is perfective.

Its infinitive is предложить. The imperfective partner is usually предлагать.

Why this matters:

• предложил = he made the offer, as a completed event
• предлагал = he was offering / used to offer / offered in an ongoing or repeated sense

In this sentence, the speaker refers to one completed action after the walk, so предложил is the natural choice.

Only the past-tense verb would need to change, and possibly the noun phrase if you changed friend itself.

For example:

• Моя подруга предложила мне чай. = My female friend offered me tea.

Changes:

• мой друг → моя подруга
• предложил → предложила

The verb agrees in gender with the subject in the past tense.

The main stress patterns are:

• послЕ
• прогУлки
• мой
• друг
• предложИл
• мне
• чай

So roughly:

ПослЕ прогУлки мой друг предложИл мне чай.

A very approximate English-style guide would be:

pas-LYE pra-GOOL-kee moy droog prad-la-ZHEEL mnye chay

But it is better to learn the real Russian sounds than rely too much on English spelling.