बैग काउंटर के बगल में रखिए।

Hindi usually follows Subject-Object-Verb order, so the verb often comes last.

Here, the understood subject is you, the object is बैग, the location is काउंटर के बगल में, and the verb रखिए comes at the end.

A very literal breakdown is:

बैग = bag
काउंटर के बगल में = next to the counter
रखिए = put/place

So this word order is normal in Hindi.

रखिए is a polite imperative form of रखना, which means to put / to keep / to place.

It is used when speaking respectfully, usually with आप.

Compare:

रख = put, very intimate or abrupt
रखो = put, informal
रखिए = put, polite
रखिएगा = put, extra polite / softer

So रखिए is the respectful way to give this instruction.

In Hindi, the subject is often left out when it is obvious from the verb form.

Because रखिए is already a polite imperative, the listener understands that the hidden subject is आप.

So:

बैग काउंटर के बगल में रखिए।
and
आप बैग kाउंटर के बगल में रखिए।

both work, but the version without आप is very natural.

Because बैग here is a direct object, and in Hindi, inanimate direct objects often appear without को.

So बैग रखिए is normal.

Using को here would sound unusual in ordinary speech unless there were some special emphasis or context. English speakers often expect an object marker everywhere, but Hindi does not work that way.

This is a very common Hindi pattern:

X के बगल में = next to X / beside X

Here:

काउंटर = counter
बगल = side
में = in / at
के = part of the linking structure in this expression

So the whole phrase literally feels something like:

at the side of the counter

You should learn के बगल में as a set location expression.

Examples:

कुर्सी मेज़ के बगल में है। = The chair is next to the table.
दुकान बैंक के बगल में है। = The shop is next to the bank.

It is an English loanword written in Devanagari script.

Hindi uses many everyday borrowed words, especially for modern objects and places. काउंटर is a normal, natural word in modern Hindi.

The same is true for बैग. It is also a common loanword.

So this sentence mixes native Hindi grammar with borrowed vocabulary, which is extremely common.

Yes, Hindi nouns normally have grammatical gender, and loanwords usually get treated as masculine or feminine by usage.

बैग is often treated as masculine by many speakers, but with some loanwords there can be variation.

In this sentence, though, you do not see any gender effect clearly, because nothing in the sentence has to agree with बैग in an obvious way.

You would notice gender more easily in sentences like:

यह बैग बड़ा है। = This bag is big.
Here बड़ा shows masculine agreement.

Yes.

Some common alternatives are:

बैग काउंटर के बगल में रखो।
Informal, used with tum

बैग काउंटर के बगल में रखिए।
Polite, standard

बैग काउंटर के बगल में रख दीजिए।
Very natural polite request, often softer than plain रखिए

बैग काउंटर के बगल में रखिएगा।
Extra polite or gentle

So if you are talking to a customer, stranger, elder, or someone you want to address respectfully, रखिए or रख दीजिए is a good choice.

Yes. Hindi word order is somewhat flexible as long as the verb stays near the end.

Both of these are natural:

बैग काउंटर के बगल में रखिए।
काउंटर के बगल में बैग रखिए।

The difference is mostly about emphasis or flow, not basic meaning.

English speakers often expect one fixed order, but Hindi allows more movement than English does.

A rough pronunciation guide is:

baig kaa-un-ter ke bagal mein rakh-yee-ay

A few points:

• बैग sounds close to English bag, though the vowel may vary a bit
• काउंटर is basically counter adapted to Hindi pronunciation
• बगल is roughly bagal
• में has a nasal sound at the end, roughly mein or mayn
• In रखिए, the ख is an aspirated sound, so say rakh, not plain rak
• रखिए is usually pronounced something like rakh-yee-ay

If you say it smoothly, the whole sentence flows naturally as:

baig kaa-un-ter ke bagal mein rakh-yee-ay