Problema încă există.

Because Romanian puts the definite article at the end of the noun.

• problemă = problem / a problem (dictionary form)
• o problemă = a problem
• problema = the problem

So Problema încă există. uses the definite form and means the problem still exists.

For this noun, the ending changes from -ă to -a in the definite singular form.

Yes. Problemă is feminine in Romanian.

That matters because feminine singular nouns often take the definite article as -a:

• o casă → casa
• o problemă → problema

So the form problema also tells you the noun is feminine singular and definite.

Here, încă means still.

So:

• Problema încă există. = The problem still exists.

In other contexts, încă can also help express ideas like more or another, especially in combinations such as încă un = one more / another. But in this sentence, before the verb, it clearly means still.

Există is the present tense form of the verb a exista (to exist).

In this sentence, it means exists:

• problema există = the problem exists

A useful thing to know is that există is used for both:

• el/ea există = he/she/it exists
• ei/ele există = they exist

Here it is singular because the subject is problema.

Because există already means exists.

English uses:

• The problem still exists

Romanian does the same idea with one verb:

• Problema încă există

You do not need este here.
Este means is, but it is used in different kinds of sentences, such as:

• Problema este gravă. = The problem is serious.

So:

• există = exists
• este = is

Because Romanian usually does not need a subject pronoun when the subject is already clear.

Here, the subject is explicitly stated:

• Problema = the problem

So adding a pronoun would be unnecessary. Romanian normally just says:

• Problema încă există.

This is very natural.

Yes. Problema există încă is also correct.

Both are natural:

• Problema încă există
• Problema există încă

The difference is mainly one of rhythm or emphasis, not basic meaning.
A learner can safely use Problema încă există, which is a very straightforward and neutral order.

Not exactly.

• Problema încă există. = The problem still exists.
  This refers to a specific, known problem.

• Încă există o problemă. = There is still a problem.
  This is more like an existential statement: a problem remains.

So the Romanian sentence with problema is more definite and specific than the English-style idea there is still a problem.

A simple learner-friendly approximation is:

• încă ≈ uhn-kuh, but with a vowel at the start that does not exist in normal English
• există ≈ eg-ZIS-tuh

A few key sounds:

• î is a specifically Romanian vowel with no exact English equivalent
• ă sounds like a short, unstressed uh
• x in există is pronounced like gz here

So the whole sentence is approximately:

• pro-BLE-ma uhn-kuh eg-ZIS-tuh

Yes, in correct standard Romanian you should write them:

• Problema încă există.

The diacritics here are:

• î
• ă

You may sometimes see informal typing without them:

• Problema inca exista.

People will usually understand it, but it is not the proper standard spelling. For learning Romanian, it is best to use the diacritics.