साबुन और तौलिया आईने के पास हैं; कंघी मेज़ पर है।

Because the first subject is compound/plural: साबुन और तौलिया = the soap and the towel.
When two nouns are joined by और (and), Hindi normally treats them as plural, so you use हैं.

The second clause has only one subject: कंघी = the comb, so it takes singular है.

• साबुन और तौलिया ... हैं = soap and towel ... are
• कंघी ... है = the comb ... is

Because आईना is a masculine noun ending in -ा, and before a postposition it usually changes to its oblique form.

So:

• direct form: आईना = mirror
• oblique form: आईने = used before a postposition

Here it is followed by के पास, so you get:

• आईने के पास = near the mirror

This is a very common pattern with masculine -ा nouns:

• कमरा → कमरे में
• लड़का → लड़के के साथ
• आईना → आईने के पास

Here के पास means near, next to, or by.

So:

• आईने के पास = near the mirror / by the mirror

A useful point: के पास is best learned as a set expression. You usually cannot drop the के and just say आईने पास.

Also, के पास can mean different things depending on context:

• location: आईने के पास = near the mirror
• possession: मेरे पास किताब है = I have a book

So in this sentence, it is clearly the location meaning.

Because मेज़ does not show a different oblique form in the singular.

In Hindi, a noun before a postposition is technically in the oblique case, but not every noun changes visibly.
Many feminine nouns and many borrowed words keep the same form.

So:

• direct: मेज़
• oblique: मेज़
• मेज़ पर = on the table

By contrast, आईना does change visibly:

• आईना → आईने के पास

They describe different kinds of location.

• के पास = near / next to / by
• पर = on / on top of / at

So:

• आईने के पास means the soap and towel are near the mirror
• मेज़ पर means the comb is on the table

In many contexts, पर suggests being on a surface or at that spot, while के पास means close to something, not necessarily touching it.

Hindi usually puts the location before the verb, and the verb comes at the end of the clause.

So the pattern is:

• [thing] + [location] + है/हैं

Examples from the sentence:

• साबुन और तौलिया
  • आईने ke पास
    • हैं
• कंघी
  • मेज़ पर
    • है

A very literal English-style breakdown would be:

• Soap and towel mirror near are
• Comb table on is

That is normal Hindi word order.

Yes, all Hindi nouns have grammatical gender.

In this sentence, the nouns are usually treated as:

• साबुन — masculine
• तौलिया — masculine
• आईना — masculine
• कंघी — feminine
• मेज़ — usually feminine

Gender does matter in Hindi, but in this sentence you do not see much of it in the verb, because है / हैं in the present tense do not change for gender.

You mainly notice grammar here through:

• number: है vs हैं
• case: आईना → आईने

Because Hindi does not use articles the way English does.

So a bare noun like कंघी can mean:

• a comb
• the comb

The exact meaning depends on context.

That is why साबुन, तौलिया, आईना, कंघी, and मेज़ appear without any word matching English a/an/the.

Yes, in everyday spoken Hindi, पे is a very common colloquial form of पर.

So these both occur:

• मेज़ पर है
• मेज़ पे है

In careful writing and in many textbooks, पर is usually preferred, but पे is completely normal in conversation.