Zdrav sam kada ne jedem mnogo šećera.

Because zdrav has to agree with the speaker.

• zdrav sam = a male speaker
• zdrava sam = a female speaker

Serbian adjectives agree in gender and number. Since the subject I is understood rather than stated, the adjective still shows whether the speaker is male or female.

Because sam is a clitic in Serbian, and clitics usually go in second position in the clause.

So these are natural:

• Zdrav sam
• Ja sam zdrav

But Sam zdrav is not normal standard word order.

This is a very common Serbian pattern:

• Umoran sam
• Srećan sam
• Spreman sam

The clitic sam comes right after the first element.

Here sam is the 1st person singular present of biti = to be.

So:

• sam = am

In this sentence, zdrav sam literally means healthy am, which Serbian uses for I am healthy.

A learner might also have seen sam meaning alone in other contexts, but that is a different word. Here it is definitely the verb to be.

It can mean either, depending on context.

Serbian present tense often covers both:

• I eat
• I am eating

In this sentence, because of the general meaning with kada, it is understood as a habitual/general action:

• kada ne jedem mnogo šećera = when I do not eat much sugar

Also, jedem is the 1st person singular present of jesti = to eat.

Because in Serbian, ne is usually written as a separate word before the verb.

So:

• jedem = I eat
• ne jedem = I do not eat

This is the normal pattern with most verbs.

A few very common forms are exceptions, such as:

• nisam
• nemam

But with jedem, the standard form is ne jedem.

Because after quantity words like mnogo, Serbian normally uses the genitive form.

So:

• šećer = sugar
• šećera = genitive form

That is why you get:

• mnogo šećera = much / a lot of sugar

This is a very common pattern:

• mnogo vode = much water
• mnogo vremena = a lot of time
• mnogo ljudi = many people

Here it is genitive singular.

That may feel strange to an English speaker, but sugar is a mass noun, so Serbian treats it as an uncountable substance here. After mnogo, that gives:

• mnogo šećera

So even though English says much sugar, not many sugars, Serbian likewise uses the mass-noun idea here.

They mean the same thing here: when.

• kada = full form
• kad = shorter, very common everyday form

So both are natural:

• Zdrav sam kada ne jedem mnogo šećera.
• Zdrav sam kad ne jedem mnogo šećera.

The shorter kad is extremely common in speech.

Not in the same way.

• kada / kad = when
• ako = if

In your sentence, the meaning is about a time or repeated situation:

• I am healthy when I do not eat much sugar

If you used ako, it would shift toward a conditional meaning:

• I am healthy if I do not eat much sugar

That is similar in English, but not exactly the same nuance.

Because jedem comes from the imperfective verb jesti, which is the normal choice for habitual, ongoing, or general actions.

This sentence describes a general habit:

• when I do not eat much sugar

The perfective pojedem would suggest a completed act, more like:

• when I eat up / consume

That does not fit as well here. For general habits, jesti / jedem is the natural choice.

In Serbian, a clause introduced by kad(a) often has no comma when it comes after the main clause and is closely connected to it.

So this is natural:

• Zdrav sam kada ne jedem mnogo šećera.

But if the kada clause comes first, then you normally use a comma:

• Kada ne jedem mnogo šećera, zdrav sam.

So the punctuation changes with the word order.

Yes, absolutely.

Serbian often omits subject pronouns because the verb form already shows the person. So:

• Zdrav sam kada ne jedem mnogo šećera.
• Ja sam zdrav kada ne jedem mnogo šećera.

Both are correct.

The version without ja is more neutral. Adding ja can give a little more emphasis, contrast, or clarity.

A rough guide:

• š sounds like sh
• ć is a soft sound, somewhat like a softer version of English ch
• šećera is roughly sheh-cheh-ra, but with a softer middle consonant than normal English ch

Also:

• šećer = sugar
• šećera = of sugar / sugar in the genitive form

For English speakers, ć is often one of the trickier Serbian sounds, so it is normal if it takes practice.