Piątego marca kończę kurs i zaczynam nową pracę.

Because Polish dates are normally expressed with:

• the day number as an ordinal in the genitive
• the month name also in the genitive

So:

• piąty = fifth
• piątego = of the fifth / on the fifth
• marzec = March
• marca = of March

So piątego marca literally works like the fifth of March, and in context it means on the fifth of March.

In Polish, you usually do not need a preposition for full dates like this.

So English on the fifth of March becomes simply:

• piątego marca

This is one of the standard ways Polish expresses dates.
By contrast:

• w marcu = in March
• piątego marca = on the fifth of March

So the lack of a preposition is normal.

Because in dates, the day is given as an ordinal numeral in the genitive case.

So:

• pierwszy → pierwszego
• drugi → drugiego
• piąty → piątego

That -ego ending is the genitive form here.

You can think of an implied word like dnia (day) behind it, which helps explain the form historically and grammatically.

Because month names also go into the genitive when used in dates.

Examples:

• styczeń → stycznia
• luty → lutego
• marzec → marca

So:

• piątego marca = the fifth of March

This is just the normal date pattern in Polish.

Because in Polish, the present tense of imperfective verbs is often used for the future, especially when talking about:

• plans
• schedules
• arrangements
• things that are definitely going to happen

So kończę kurs i zaczynam nową pracę can mean:

• I’m finishing the course and starting a new job
• or, in natural English, I finish the course and start a new job
• or more idiomatically, I’m finishing the course and starting a new job on March 5th

This is very common in Polish.

Yes, they could be.

Compare:

• kończę / zaczynam = imperfective
• skończę / zacznę = perfective

So:

• Piątego marca kończę kurs i zaczynam nową pracę.
• Piątego marca skończę kurs i zacznę nową pracę.

Both can refer to the future, but the feeling is slightly different:

• imperfective can sound more like a planned event or scheduled action
• perfective focuses more on the fact that the actions will be completed / initiated as single whole events

In everyday speech, both are possible depending on nuance.

Because both words are direct objects, so they are in the accusative case. But different noun types behave differently in the accusative.

• kurs is masculine inanimate
• praca is feminine

For masculine inanimate nouns, the accusative singular is usually the same as the nominative:

• kurs → kurs

For many feminine nouns ending in -a, the accusative singular changes to -ę:

• praca → pracę

So:

• kończę kurs
• zaczynam pracę

Because the adjective has to match the noun in:

• gender
• number
• case

Here, pracę is:

• feminine
• singular
• accusative

So the adjective nowy must also be in the feminine singular accusative:

• nowa praca = nominative
• nową pracę = accusative

That is why both words change together.

No, Polish word order is fairly flexible.

The sentence as given:

• Piątego marca kończę kurs i zaczynam nową pracę.

is very natural because it puts the date first, which sets the time frame right away.

You could also say:

• Kończę kurs i zaczynam nową pracę piątego marca.

That is also grammatical, but it places the date later and may sound slightly different in emphasis.

So the original version is a natural choice if you want to highlight when this happens.

Yes, the most natural reading is that both actions are tied to piątego marca.

So it means that on March 5th:

• I finish the course
• and I start a new job

If you wanted to make that connection even more explicit, Polish could do that in other ways too, but in this sentence the date normally applies to both verbs.

Because in Polish, month names are not capitalized unless they begin a sentence.

So:

• marzec
• marca

both normally take lowercase.

This is different from English, where March is always capitalized.

A rough guide is:

• koń- sounds like koń with a soft ń
• -czę sounds like cheh with Polish cz

A very approximate English-friendly version might be:

• KON-che

But that is only approximate, because Polish has sounds English does not match exactly.

A few key points:

• ń is a soft n sound
• cz sounds like ch in church
• ę is a nasal vowel, though in normal speech its exact pronunciation depends on the sound that follows

So it is best learned by listening, but KON-che is a useful first approximation.