У этой розы мягкие лепестки.

Russian very often expresses possession with у + genitive. So the structure is literally closer to At this rose, there are soft petals, but the natural English meaning is This rose has soft petals.

This pattern is extremely common:

• У меня есть книга. = I have a book.
• У этой розы мягкие лепестки. = This rose has soft petals.

So even though English often uses ’s or has, Russian often uses у.

Because у requires the genitive case.

The base forms are:

• эта = this (feminine)
• роза = rose

In the genitive singular, they become:

• эта → этой
• роза → розы

So:

• у этой розы = of/at this rose, which is how Russian builds the possession structure here.

Because роза has to change to the genitive singular after у.

• роза = nominative singular
• розы = genitive singular

So у этой розы is correct because у governs the genitive.

By the word этой.

Этой is feminine singular genitive, so it tells you that розы here means of this rose, not roses.

If it were plural, you would get something like:

• у этих роз = of these roses

So the form of the demonstrative helps you identify the number.

Because лепестки is plural nominative, and the adjective мягкие agrees with it.

Agreement in Russian means the adjective matches the noun in:

• number
• gender
• case

Here:

• лепестки = plural nominative
• мягкие = plural nominative

In this kind of affirmative possession sentence, the thing being possessed is usually in the nominative:

• У розы мягкие лепестки.

That is why you do not see genitive on лепестки here.

Yes, in a sense. Russian often leaves out the present-tense to be.

So the sentence can be understood as something like:

• У этой розы есть мягкие лепестки.

But in natural Russian, есть is often omitted, especially in simple descriptive statements.

Compare:

• У этой розы мягкие лепестки. = neutral, descriptive
• У этой розы есть мягкие лепестки. = more explicit about possession/existence

Both are possible, but the version without есть is very normal.

Yes, absolutely.

The version with есть is grammatically correct and natural. It can sound a little more like you are stating the existence of the petals as a feature the rose has.

Very roughly:

• У этой розы мягкие лепестки. = descriptive, matter-of-fact
• У этой розы есть мягкие лепестки. = slightly more explicit: this rose does have soft petals

In many everyday situations, either version works.

That is the most neutral word order for this idea. Russian often puts the possessor first in this kind of sentence.

So:

• У этой розы мягкие лепестки. = neutral

But Russian word order is flexible, and changing it changes emphasis:

• Мягкие лепестки у этой розы. = emphasis on soft petals
• Лепестки у этой розы мягкие. = emphasis on the petals are soft

So the original order is the safest and most standard one.

Russian does not have articles like a and the.

Whether something is definite or indefinite is usually understood from:

• context
• word order
• demonstratives like этот / эта / это

Here, этой already gives a clear meaning: this rose. So Russian does not need an article.

The stress is:

У э́той ро́зы мя́гкие лепестки́.

A rough pronunciation guide:

• у = oo
• э́той ≈ EH-tay
• ро́зы ≈ ROH-zih
• мя́гкие ≈ MYAK-kee-ye
• лепестки́ ≈ lee-pyes-TKEE

The most important stress points are:

• э́той
• ро́зы
• мя́гкие
• лепестки́

Yes. You could say:

У розы мягкие лепестки.

That would mean something like The rose has soft petals or A rose has soft petals, depending on context.

Adding этой makes it specifically this rose. So этой is there to point to a particular rose.