Ben çayın yanında biraz kraker yiyorum.

Yes. In Turkish, the ending -um in yiyorum already shows that the subject is I, so ben is often optional.

• Yiyorum = I am eating
• Ben yiyorum = I am eating

Including ben can add:

• emphasis,
• contrast,
• or simple clarity in context.

So this sentence could also be said as:

• Çayın yanında biraz kraker yiyorum.

That would still be perfectly natural.

Because yanında usually works with a noun in the genitive form.

Here is the pattern:

• çay = tea
• çayın = of the tea / tea’s
• yanında = at its side / next to it

So:

• çayın yanında = at the side of the tea, which in natural English can mean with the tea or alongside the tea

This structure is very common:

• evin yanında = next to the house
• okulun yanında = next to the school
• kahvenin yanında = with the coffee / next to the coffee

Literally, yanında means at its side or next to it.

It comes from:

• yan = side
• -ı = possessive element
• -nda = locative ending (in/on/at)

So çayın yanında literally means something like at the tea’s side.

In food contexts, Turkish often uses this expression in a more natural English sense:

• kahvenin yanında çikolata = chocolate with coffee
• çayın yanında kraker = crackers with tea

So the literal image is beside the tea, but the practical meaning is often served with or eaten with.

After quantity expressions like biraz, Turkish usually does not pluralize the noun.

So:

• biraz kraker = some crackers
• not biraz krakerler

This is different from English, where we usually say some crackers with a plural noun.

Turkish often keeps the noun in a basic singular-looking form after words showing quantity:

• iki kitap = two books
• birkaç arkadaş = a few friends
• biraz ekmek = some bread
• biraz kraker = some crackers

So even though the English translation may be plural, Turkish often does not use -ler / -lar there.

Because it is an indefinite direct object.

In Turkish, a direct object often takes the accusative ending only when it is:

• specific,
• definite,
• or clearly identified.

Compare:

• Biraz kraker yiyorum. = I am eating some crackers.

  • indefinite, non-specific

• Krakeri yiyorum. = I am eating the cracker / the crackers.

  • specific, definite

So in your sentence, kraker is just some crackers, not a specific known set of crackers, so no accusative ending is used.

Yiyorum comes from the verb yemek, which means to eat.

Breakdown:

• ye- = verb stem
• -iyor- = present continuous marker
• -um = I

So:

• ye + iyor + um → yiyorum

It means:

• I am eating
• or sometimes just I eat, depending on context, though the main feel is present continuous

Some useful related forms:

• yiyorsun = you are eating
• yiyor = he/she/it is eating
• yiyoruz = we are eating
• yiyorsunuz = you are eating
• yiyorlar = they are eating

Because Turkish normally prefers Subject-Object-Verb word order, and the verb often comes last.

So this sentence follows a very typical Turkish pattern:

• Ben = subject
• çayın yanında = adverbial phrase
• biraz kraker = object
• yiyorum = verb

Turkish word order is flexible, but the verb very often stays at the end.

For example, these can all be possible depending on emphasis:

• Ben çayın yanında biraz kraker yiyorum.
• Çayın yanında biraz kraker yiyorum.
• Biraz krakeri çayın yanında yiyorum. (with a different nuance if accusative is used)

The original version sounds neutral and natural.

Yes, often you could, but the nuance is slightly different.

• çayla = with tea
• çayın yanında = at the side of the tea / with the tea / alongside tea

In many everyday situations, both can work.

Compare:

• Çayla biraz kraker yiyorum. = I’m eating some crackers with tea.
• Çayın yanında biraz kraker yiyorum. = I’m eating some crackers with the tea / alongside the tea.

Çayın yanında can sound a little more like served next to or as an accompaniment to the tea.
Çayla is a more direct with tea.

Biraz literally means a little or some, and it is very common with uncountable nouns, but in everyday Turkish it can also be used for a small indefinite amount of countable food items.

So:

• biraz su = some water
• biraz ekmek = some bread
• biraz kraker = some crackers

With foods and snacks, this is quite natural. The idea is not to count them one by one, but to refer to a small amount in general.

If you wanted to be more explicitly countable, you might also hear:

• birkaç kraker = a few crackers

So biraz kraker is natural when you just mean some crackers in a loose, everyday way.