Learn Turkish: Burada stajyer olarak çalışıyorum.

Yes! It is made by adding the suffix -(y)arak/-(y)erek (which means "by doing" or "while doing") to the verb root ol-. So stajyer olarak literally means "being an intern." In English, we naturally translate this as "as an intern."

Does the noun before olarak need any specific case ending?

No. When used to mean "as" or "in the role of," olarak directly follows the plain, dictionary form of the noun. That's why it is simply stajyer olarak, with no extra case suffixes on stajyer.

The English translation says "I work", but çalışıyorum is present continuous ("I am working"). Why?

In Turkish, the present continuous tense (-iyor) is standard for stating where you currently work, even if it's a general, ongoing fact rather than an action happening this exact second. English often prefers the simple present ("I work") for this, but Turkish prefers -iyor.

Stajyer (intern) has an unusual spelling. How is the j pronounced here?

It is a French loanword. In Turkish, the letter j is always pronounced like the "s" in the English word "measure" or "vision" (a soft "zh" sound). It is relatively rare and mostly found in foreign loanwords.