Stomatologul programează radiografia dentară.

Because Romanian usually puts the definite article on the end of the noun, instead of using a separate word like English the.

• stomatolog = dentist
• stomatologul = the dentist

For a masculine singular noun like this one, -ul is a very common definite ending.

Because radiografia is the definite form: the X-ray / the radiograph.

• radiografie = an X-ray / X-ray
• radiografia = the X-ray

So this sentence is talking about a specific dental X-ray, not just any one. If you wanted an indefinite meaning, you could say o radiografie dentară.

In Romanian, adjectives often come after the noun. So radiografie dentară is the normal word order for dental X-ray.

• radiografie = noun
• dentară = adjective

This is very common in Romanian:

• cabinet stomatologic
• problemă medicală
• radiografie dentară

English usually puts the adjective before the noun, but Romanian often puts it after.

Because it has to agree with radiografia in gender and number.

radiografie / radiografia is feminine singular, so the adjective also has to be feminine singular:

• masculine singular: dentar
• feminine singular: dentară
• masculine plural: dentari
• feminine plural: dentare

So radiografia dentară is correct because both words match.

In this normal noun + adjective pattern, Romanian usually marks definiteness on the noun, not separately on the adjective.

So:

• radiografie dentară = a dental X-ray
• radiografia dentară = the dental X-ray

The noun carries the definite article, and the adjective simply agrees with it.

Yes. A programa can mean to schedule, to book, or to arrange an appointment for, depending on context.

In medical or administrative contexts, it very often means to schedule:

• a programa o consultație = to schedule an appointment
• a programa o radiografie = to schedule an X-ray

So this is a useful “false friend” to notice: it does not only mean to program in the computer/software sense.

It is the present tense, 3rd person form of a programa.

Here it means:

• he/she schedules
• or they schedule

Romanian often uses the same form for he/she and they in the present tense for this type of verb. In this sentence, it must be singular because the subject is stomatologul = the dentist.

So here:

• stomatologul programează = the dentist schedules

Because pe is usually used with specific animate direct objects, especially people.

For example:

• Stomatologul îl programează pe pacient. = The dentist schedules the patient.

But radiografia dentară is an inanimate thing, so no pe is needed:

• Stomatologul programează radiografia dentară.

Yes, in many situations you can.

• stomatolog is a standard medical/formal word
• dentist also exists and is very common in everyday speech

So Stomatologul programează radiografia dentară and Dentistul programează radiografia dentară are both understandable. The first can sound a bit more formal or technical.

A rough pronunciation is:

pro-gra-MEA-ză

A few helpful points:

• ă is a separate Romanian vowel, pronounced like the a in English sofa
• ea is pronounced together as one vowel group
• the stress falls on MEA

So the ending -ează is very common in Romanian verbs and is worth getting used to.

Yes, Romanian word order is more flexible than English, although Subject + Verb + Object is the most neutral order here:

• Stomatologul programează radiografia dentară.

That is the most straightforward version.

Other orders are possible for emphasis, but they may sound more marked or require extra words. So for a learner, this sentence is the best basic pattern to remember.