Lekar misli da moram da pijem mnogo vode.

They are doing two different jobs.

• Lekar misli da... = The doctor thinks that...
  • Here da introduces a content clause, like English that.
• moram da pijem = I have to drink
  • Here da is part of a very common Serbian pattern: modal verb + da + present tense.

So the sentence is built like this:

• Lekar misli da = The doctor thinks that...
• moram da pijem mnogo vode = I have to drink a lot of water.

In Serbian, after verbs like morati (must / have to), it is very common to use:

• moram da pijem
• literally: I must that I drink
• naturally: I have to drink

This is one of the biggest differences from English. English uses the infinitive (to drink), but Serbian very often uses da + present tense instead.

So:

• moram da pijem = I have to drink
• not usually moram piti in everyday Serbian

You may sometimes see an infinitive in related languages or in more formal/literary usage, but da + present is the normal Serbian pattern.

Yes, pijem by itself means I drink / I am drinking.

But after da, Serbian uses a present-tense form even when English would use an infinitive.

So:

• pijem = I drink
• da pijem = that I drink / to drink, depending on context

In this sentence:

• moram da pijem = I have to drink

So even though the form is present tense, the whole phrase functions like English have to drink.

Serbian often drops subject pronouns when the verb already makes the subject clear.

• moram already means I must / I have to
• so ja is not necessary

You could say:

• Lekar misli da ja moram da pijem mnogo vode

But that sounds more emphatic, as if you want to stress I specifically.

The natural neutral version is:

• Lekar misli da moram da pijem mnogo vode.

Because mnogo (a lot of / much) requires the noun after it to be in the genitive.

So:

• voda = water
• mnogo vode = a lot of water

This is very important in Serbian:

• čaša vode = a glass of water
• mnogo vode = a lot of water

Since water is an uncountable/mass noun, Serbian uses genitive singular here:

• nominative: voda
• genitive: vode

Mnogo means much or a lot of.

In this sentence:

• mnogo vode = a lot of water

It works as a quantity word. After it, the following noun usually goes into the genitive:

• mnogo vode = a lot of water
• mnogo knjiga = many books

So the pattern is:

• mnogo + genitive

Misli is the 3rd person singular present tense of misliti (to think).

So:

• ja mislim = I think
• ti misliš = you think
• on/ona misli = he/she thinks

Since lekar is singular, you use:

• Lekar misli = The doctor thinks

Because Serbian uses the present tense for both verbs here in a perfectly normal way.

• Lekar misli = The doctor thinks
• moram da pijem = I have to drink

This expresses a current opinion and a current obligation/necessity.

So the sentence means something like:

• The doctor thinks that I need to drink a lot of water now / in general.

No future tense is needed unless you specifically want a future meaning.

Yes. Both are used for doctor.

• lekar = doctor, physician
• doktor = doctor

In everyday speech, doktor is extremely common. Lekar is also completely normal and standard.

So both of these work:

• Lekar misli da moram da pijem mnogo vode.
• Doktor misli da moram da pijem mnogo vode.

A small nuance:

• doktor can also mean someone with a doctorate, depending on context
• lekar refers specifically to a medical doctor

The given word order is the neutral, natural one:

• Lekar misli da moram da pijem mnogo vode.

Serbian word order is fairly flexible, but changing it can add emphasis.

For example:

• Lekar misli da moram da pijem mnogo vode.
  Neutral: The doctor thinks I have to drink a lot of water.

• Mnogo vode moram da pijem, lekar misli.
  More marked/emphatic, less neutral.

So yes, other orders are possible, but the original sentence is the best basic version for a learner to model.

It can correspond to both, depending on context.

• moram = I must / I have to

In this sentence, English usually sounds more natural with:

• The doctor thinks that I have to drink a lot of water.

But must is also possible:

• The doctor thinks that I must drink a lot of water.

So morati covers both ideas of necessity.

In real-life translation, yes, it can sound similar, but grammatically moram is stronger than should.

• moram = must / have to
• treba da pijem = I should / I need to drink

So if you want the softer meaning should, Serbian would more naturally use:

• Lekar misli da treba da pijem mnogo vode.

Your original sentence is stronger:

• Lekar misli da moram da pijem mnogo vode. = The doctor thinks I have to / must drink a lot of water.