Без интернета программа не работает нормально.

Because без (without) requires the genitive case.

• интернет → nominative singular
• интернета → genitive singular

So:

• без интернета = without internet

This is a very common pattern in Russian:

• без воды = without water
• без денег = without money
• без помощи = without help

Не is the standard negation word in Russian, and it usually goes directly before the verb.

• работает = works / is working / functions
• не работает = does not work / isn’t working

So in this sentence:

• программа не работает = the program doesn’t work

Yes, but here работает is best understood as functions, runs properly, or operates.

Russian often uses работать (to work) for machines, devices, software, systems, and services:

• Телефон не работает. = The phone isn’t working.
• Сайт работает. = The website works / is up.
• Программа работает нормально. = The program works normally.

So работает is not about a person having a job here. It means the software is functioning.

Because программа is the subject of the sentence — it is the thing doing the action of работает.

• программа = the program / software program
• работает = works

So the basic structure is:

• Программа работает. = The program works.

Subjects are normally in the nominative case.

Нормально is an adverb meaning normally, properly, or fine depending on context.

Here it modifies the verb работает:

• работает нормально = works normally / works properly

So the sentence means that without internet, the program does not function properly.

A useful comparison:

• нормальный = normal (adjective)
• нормально = normally / properly (adverb)

Examples:

• Это нормальная программа. = This is a normal/regular program.
• Программа работает нормально. = The program works properly.

Yes, Russian word order is relatively flexible.

This sentence starts with Без интернета to set the condition first:

• Без интернета = without internet
• программа не работает нормально = the program does not work properly

This is very natural, because it highlights the circumstance first.

Other possible word orders include:

• Программа без интернета не работает нормально.
• Программа не работает нормально без интернета.

These all mean roughly the same thing, but the emphasis changes slightly.

• Без интернета... emphasizes the condition right away.
• Putting it later sounds a bit more neutral or less focused on that condition.

Usually just internet in a general sense, as in English:

• without internet
• without an internet connection

Russian does not use articles like a or the, so интернета by itself does not tell you whether it is the internet or simply internet access. Context decides.

In this sentence, без интернета most naturally means:

• without internet access
• when there is no internet connection

Most likely, yes.

Программа can mean several things depending on context:

• a software program
• a TV/radio program
• a schedule or plan
• a program of study

But because the sentence says Без интернета программа не работает нормально, the most natural meaning is software/application.

So here программа is best understood as:

• program
• app
• software

Работает comes from the imperfective verb работать, which is the normal choice for describing a general state, repeated behavior, or ongoing functioning.

Here the sentence describes a general fact:

• Without internet, the program does not function properly.

That is exactly the kind of meaning where imperfective is used.

A perfective verb would not fit naturally here, because perfective usually refers to a completed action or a single result, not a general state of functioning.

The sentence specifically says:

• не работает нормально = doesn’t work normally / properly

So it does not necessarily mean the program completely fails to run. It could mean:

• some features stop working
• performance becomes limited
• it behaves incorrectly
• it is unusable in practice

If you wanted to say it does not work at all, Russian might say:

• Без интернета программа не работает. = Without internet, the program doesn’t work.

Adding нормально softens or narrows the statement to not properly / not as it should.

Normally, no.

In Russian, интернет is usually written with a lowercase letter in modern standard usage:

• интернет
• без интернета

Older texts or some stylistic choices may use Интернет, but lowercase is now the usual form.

So in most modern writing, the sentence should be:

• Без интернета программа не работает нормально.