Ovaj kaput mi se ne sviđa; da li mogu da probam sledeći?

Because ovaj kaput is the subject of the first clause, so both words are in the nominative singular masculine.

• ovaj = this (masculine nominative singular)
• kaput = coat (masculine nominative singular)

The verb phrase sviđa se works differently from English: the thing that is liked is actually the grammatical subject.

So Serbian literally works more like:

• This coat is pleasing to me.

That is why ovaj kaput stays in the nominative.

Because sviđati se / svideti se is structured differently from English.

In Serbian:

• the person who likes something goes in the dative
• the thing liked is the subject
• se is part of the verb expression

So:

• Ovaj kaput mi se sviđa. = literally This coat pleases me = natural English I like this coat

Here:

• mi = to me
• se sviđa = is pleasing / appeals

This is one of the most important patterns to get used to in Serbian.

Mi is the unstressed dative form of ja (I / me), and here it means to me.

So:

• mi se sviđa = is pleasing to me
• ti se sviđa = is pleasing to you
• mu se sviđa = is pleasing to him
• joj se sviđa = is pleasing to her

Examples:

• Ova haljina mi se sviđa. = I like this dress.
• Ova haljina ti se sviđa? = Do you like this dress?

Se is a reflexive particle that belongs with this verb expression. You generally learn it as part of the verb:

• sviđati se = to be pleasing, to appeal
• svideti se = to come to be liked / to please

So you should think of sviđa se as a unit.

You normally cannot just remove se and keep the same meaning.
In this sentence:

• mi se sviđa = I like it / it pleases me

Because the subject is singular:

• ovaj kaput = this coat → singular
• therefore: sviđa se → singular verb form

Compare:

• Ovaj kaput mi se sviđa. = I like this coat.
• Ovi kaputi mi se sviđaju. = I like these coats.

So the verb agrees with the thing being liked, not with the person who likes it.

This is standard Serbian clitic order.

The short unstressed words (mi, se) usually come early in the clause, and ne goes directly with the verb:

• mi = dative clitic
• se = reflexive clitic
• ne sviđa = negated verb

So:

• Ovaj kaput mi se ne sviđa.

is the natural order.

A learner does not need to memorize every clitic rule immediately, but it helps to recognize that mi se is a very common sequence.

Yes. That is also correct and very natural.

Both are fine:

• Ovaj kaput mi se ne sviđa.
• Ne sviđa mi se ovaj kaput.

The difference is mainly emphasis:

• Ovaj kaput mi se ne sviđa puts a bit more focus on this coat
• Ne sviđa mi se ovaj kaput starts with the negative statement itself: I don't like this coat

Serbian word order is more flexible than English, especially for emphasis and style.

Da li is a common way to form a yes/no question.

So:

• Da li mogu da probam sledeći? = Can I try the next one?

Another very common equivalent is:

• Mogu li da probam sledeći?

Both are correct.
The da li version is often very clear and easy for learners to use.

They do two different jobs.

1. da li = question marker

   • Da li mogu...? = Can I...?

2. the second da introduces the verb after mogu

   • mogu da probam = I can try

So the structure is:

• da li = whether / question marker
• mogu da probam = I can try

This double da is completely normal in Serbian.

In modern Serbian, after modal verbs like moći, it is very common to use:

• present tense + da

So:

• mogu da probam = I can try

You may also hear or see the infinitive in some contexts:

• mogu probati

But mogu da probam is extremely standard and very common in everyday Serbian.

Probam is the 1st person singular present form of probati:

• ja probam = I try
• ti probaš = you try
• on/ona/ono proba = he/she/it tries

After mogu da, Serbian uses the present form:

• mogu da probam = I can try

So even though English uses to try, Serbian uses a finite verb form after da.

Sledeći means next.

In this sentence it stands by itself, but it really means:

• sledeći kaput = the next coat

Since kaput is already understood from the context, Serbian can omit it:

• da probam sledeći? = to try the next one?

So sledeći is functioning like an adjective used on its own, with the noun understood.

Because Serbian can omit the noun when it is obvious from context.

You are already talking about coats, so:

• sledeći = the next one
• full version: sledeći kaput = the next coat

Both are possible, but the shorter version sounds natural because the noun does not need to be repeated.

Because the omitted noun is kaput, which is masculine.

So the adjective must agree with the implied noun:

• sledeći (kaput) → masculine
• sledeća (haljina) → feminine
• sledeće (odelo) → neuter

Examples:

• Mogu li da probam sledeći? = the next coat
• Mogu li da probam sledeću? = the next dress
• Mogu li da probam sledeće? = the next suit / the next one (neuter)

Yes. This is an aspect difference.

• sviđati se = imperfective
• svideti se = perfective

In many everyday sentences:

• Ovaj kaput mi se ne sviđa. means I don't like this coat / This coat doesn't appeal to me

A perfective version is often used when something ends up pleasing someone or is liked upon seeing/trying it, for example:

• Ovaj kaput mi se svideo. = I liked this coat.
• Da li ti se svidela haljina? = Did you like the dress?

For your sentence, ne sviđa mi se is the natural choice.

Yes, absolutely. Both are correct.

• Da li mogu da probam sledeći?
• Mogu li da probam sledeći?

The second version can sound a bit more compact.
Both are polite and normal.

Yes, it is perfectly natural and polite.

• Ovaj kaput mi se ne sviđa; da li mogu da probam sledeći?

If you want to sound even softer or more polite, you could also say:

• Ovaj kaput mi se ne sviđa. Da li mogu da probam sledeći?
• Ovaj kaput mi se baš ne sviđa. Mogu li da probam sledeći?
• Da li bih mogao/mogla da probam sledeći?
  = Could I try the next one?

That last version is especially polite:

• mogao if the speaker is male
• mogla if the speaker is female

The semicolon simply links two closely related clauses:

• Ovaj kaput mi se ne sviđa
• da li mogu da probam sledeći?

In normal conversation, people would usually just say it as two spoken chunks, and in writing you could also use a period or even a comma depending on style:

• Ovaj kaput mi se ne sviđa. Da li mogu da probam sledeći?

That may look more natural for many learners.