Весной мама сажает цветы на клумбе.

Because Russian often uses the instrumental case to talk about seasons when meaning in spring / during spring.

• весна = spring (dictionary form, nominative)
• весной = in spring / during spring

This is a very common pattern:

• зимой = in winter
• весной = in spring
• летом = in summer
• осенью = in autumn

So Весной мама сажает цветы на клумбе means In spring, mom plants flowers in the flowerbed.

Because мама is the subject of the sentence — the person doing the action.

In this sentence:

• мама = the one who plants
• сажает = plants

So мама stays in the nominative case.

сажает is:

• 3rd person singular
• present tense
• from the verb сажать = to plant
• imperfective aspect

It agrees with мама, which is she:

• я сажаю = I plant
• ты сажаешь = you plant
• она сажает = she plants

So мама сажает = mom plants.

Because the sentence describes a habitual / repeated action, not one single completed event.

Весной мама сажает цветы... suggests something like:

• In spring, mom plants flowers
• Every spring, mom plants flowers

For repeated or general actions, Russian normally uses the imperfective.

If you wanted to stress a single completed future action, you might use a perfective verb instead, for example:

• Весной мама посадит цветы на клумбе.
  = In spring, mom will plant flowers in the flowerbed.

цветы is the accusative plural here, because it is the direct object — the thing being planted.

The dictionary singular is:

• цветок = flower

Plural nominative:

• цветы = flowers

In this sentence, we need the accusative plural, but for inanimate plural nouns, the accusative plural is the same as the nominative plural. So:

• nominative plural: цветы
• accusative plural: цветы

That is why the form does not change.

Because на клумбе expresses location, not movement.

Russian uses:

• на + prepositional for location = on/in/at
• на + accusative for motion toward a place

So:

• на клумбе = in/on the flowerbed (location)
• на клумбу = onto/to the flowerbed (movement toward it)

In this sentence, the flowers are being planted in the flowerbed, so the location meaning is intended, hence:

• на клумбе

клумбе is the prepositional singular of клумба.

• клумба = flowerbed
• на клумбе = in/on the flowerbed

It is in the prepositional case because it follows на in a location meaning.

Russian word order is flexible, and placing Весной first gives the sentence a natural time-setting feel:

• Весной мама сажает цветы на клумбе. = In spring, mom plants flowers in the flowerbed.

This is very normal in Russian. It sets the scene first.

Other word orders are possible, for example:

• Мама весной сажает цветы на клумбе.
• Мама сажает цветы на клумбе весной.

These are all understandable, but the original sounds very natural because it starts with the time expression.

No. Russian has no articles.

So words like:

• мама
• цветы
• клумбе

do not automatically show a / the.

Depending on context, the sentence could be understood as:

• Mom plants flowers in the flowerbed in spring
• Mom plants the flowers in the flowerbed in spring

Usually context tells you which meaning is intended.

The stress is:

• веснóй
• мáма
• сажáет
• цветы́
• на клýмбе

A simple pronunciation guide:

• веснóй ≈ vye-SNOY
• мáма ≈ MA-ma
• сажáет ≈ sa-ZHA-yet
• цветы́ ≈ tsvye-TY
• на клýмбе ≈ na KLYUM-bye

Stress is important in Russian, so it is worth learning these together with the words.

Yes, весной can match either in spring or in the spring in English.

Russian does not make that article distinction, so the same word covers both ideas. In most contexts:

• Весной = in spring / in the spring / during springtime

All are good natural translations depending on the style of English you want.