Не те щоб кошик важкий, але буряка забагато.

It is an idiomatic softening expression. In this sentence, Не те щоб X, але Y means something like:

It’s not exactly that X, but rather Y
or
It’s not that X so much as Y

So Не те щоб кошик важкий, але буряка забагато means the problem is not really the basket being heavy in itself; the real point is that there is too much beet/beetroot in it.

Here те is part of a fixed expression: не те щоб.

On its own, те is the neuter form of той (that), but in this phrase you should mostly learn the whole chunk together. Native speakers do not usually think of it word by word here; it functions as an idiom.

So for a learner, it is best to memorize:

не те щоб... = not exactly that... / it’s not really that...

Not in this sentence.

Very often щоб does mean so that / in order that, but here it is part of the fixed phrase не те щоб. So it is not expressing purpose.

Compare:

• Я прийшов, щоб допомогти. = I came to help.
• Не те щоб я втомився... = It’s not exactly that I’m tired...

In your sentence, щоб simply helps form the idiomatic structure.

Because кошик is a masculine singular noun in the nominative case, and важкий agrees with it.

So:

• кошик = masculine singular nominative
• важкий = masculine singular nominative adjective

This is normal adjective agreement in Ukrainian.

Also, Ukrainian usually leaves out the verb to be in the present tense, so:

• кошик важкий literally = basket heavy
• natural English = the basket is heavy

Because Ukrainian often omits є (is/are) in present-tense statements, especially in short, natural speech.

So:

• буряка забагато literally looks like of beet too much
• natural meaning is there is too much beet / there are too many beets

This kind of structure is very common with words like:

• багато = a lot
• мало = little / few
• достатньо = enough
• забагато = too much / too many

Because quantity words like багато, мало, забагато typically require the noun to be in the genitive.

So:

• буряк = nominative singular
• буряка = genitive singular

This is a very common pattern in Ukrainian:

• багато часу = a lot of time
• мало грошей = little money
• забагато роботи = too much work

So буряка забагато follows the same grammar.

This is a nuance question.

буряка is genitive singular, and here it gives a more mass/collective sense, like too much beet/beetroot or too much beet produce.

буряків is genitive plural and would emphasize separate individual beets: too many beets.

In real usage, both can be possible depending on what exactly the speaker wants to emphasize:

• буряка забагато = too much beet / too much beetroot / too much beet stuff
• буряків забагато = too many beets as individual items

The sentence you gave sounds natural and idiomatic as a general statement about excess quantity.

Забагато means too much or too many, with the idea of excess.

It is written as one word.

Compare:

• багато = a lot / many / much
• дуже багато = very much / very many
• забагато = too much / too many

So забагато is stronger than just a lot. It means the amount is more than wanted, needed, or reasonable.

Yes. Ukrainian word order is fairly flexible.

Your sentence has:

але буряка забагато

This puts буряка first, which can make it sound like as for the beet, there’s too much of it.

You could also say:

але забагато буряка

That is also natural. It may sound a bit more neutral in some contexts.

Both are grammatical; the difference is mostly in emphasis and information flow.

Because але joins two clauses, so a comma before it is normal.

The structure is:

Не те щоб кошик важкий, але буряка забагато.

There is no comma after щоб because щоб кошик важкий stays together as part of the clause introduced by не те щоб.

So the punctuation reflects the larger structure:

• first idea: Не те щоб кошик важкий
• contrast: але
• second idea: буряка забагато