Learn Hardcore Chinese: wǒmen yīnggāi duō zuò liànxí.

a recommendation or duty:
- 我们应该多做练习。 = We should do more practice.
sometimes a reasonable expectation:
- 他现在应该到家了。 = He should be home by now.

In your sentence it is clearly advice / recommendation, not a strict command like must. If you wanted something stronger, you might use 必须 (bìxū):

我们必须多做练习。 = We must do more practice.

What exactly is 多 (duō) doing in this sentence? Is it many, much, or more?

Here 多 is an adverb meaning more (in quantity or frequency).

我们应该多做练习。 = We should do more practice. / We should practice more.

It does not mean many as an adjective (like many exercises) in this position. When 多 is directly before the verb (做), it modifies the action:

多做 = do (something) more (often / more in amount)

Why is the word order 多做练习 and not 做多练习?

In Mandarin, adverbs like 多 usually go before the verb they modify:

多做练习 = do more practice (do more of the action)
多吃蔬菜 = eat more vegetables

做多练习 is not natural.
If you want many / a lot of exercises as a noun phrase, you’d normally say:

做很多练习 = do many exercises
做更多练习 = do more exercises

So:

多做练习: focuses on doing the action more.
做很多 / 更多练习: focuses on the amount of exercises as a noun.

What is the difference between 多做练习 and 多练习?

Both are correct but slightly different:

多做练习
- 练习 is a noun: practice / exercises
- 做练习 = do exercises
- 多做练习 = do more exercises
多练习
- 练习 is a verb: to practice
- 多练习 = practice more

In meaning, 我们应该多做练习 and 我们应该多练习 are very close.

多做练习 sounds a bit like do more practice tasks / exercises (discrete items, e.g., workbook exercises).
多练习 is more general practice more (in general).

Could the subject 我们 (wǒmen, we) be dropped? Is 应该多做练习 also okay?

Yes, in the right context the subject can be omitted:

(我们) 应该多做练习。

If it’s already clear from context that you’re talking about us / we, people often just say:

应该多做练习。
= (We) should do more practice.

Chinese frequently drops obvious subjects and objects when they can be inferred from context. But in isolation (like in a textbook sentence), 我们 is kept to make it clear.

Why isn’t there any word showing tense, like past or future? How do we know when this happens?

Mandarin normally does not mark tense (past / present / future) with verb endings like English does. Instead, time is understood from:

context
time expressions (e.g. 昨天, 明天, 以后)
aspect particles like 了, 过, 着

Here, 我们应该多做练习 is a general recommendation, so it’s like:

We should do more practice (now / from now on / generally).

If you wanted to make it clearly about the future, you could add:

以后我们应该多做练习。
= From now on / In the future, we should do more practice.

What is the difference between 我们应该多做练习 and 我们应该做更多练习?

Both can be translated as We should do more practice, but the structure differs:

我们应该多做练习
- 多 is an adverb modifying 做:
  - 多做 = do (it) more
- Emphasis: increase how much we do this action.
我们应该做更多练习
- 更多 is an adjective modifying the noun 练习:
  - 更多练习 = more exercises
- Emphasis: a larger number/amount of exercises.

In everyday speech, the difference is subtle; both are common and natural.

If I want to say We shouldn’t do more practice, where does 不 (bù) go?

Negation goes before the modal verb 应该:

我们不应该多做练习。
= We shouldn’t do more practice.

This means It’s not advisable / It’s wrong for us to do more practice.

If you say:

我们应该不多做练习。

that becomes very odd or changes meaning toward:

We should not do much practice (we should keep it to a small amount) — and even so, it sounds unnatural. The natural way is 我们不应该多做练习 or 我们不需要多做练习 (we don’t need to do much practice).

How strong or polite is 应该? Does it sound like a command?

应该 is softer than a command, but stronger than a casual suggestion:

It suggests duty / right thing to do / sensible choice.
In English, it sits close to should / ought to.

For comparison:

我们应该多做练习。
= We should do more practice. (reasonable advice)
我们最好多做练习。
= We had better do more practice. (strong suggestion)
我们必须多做练习。
= We must do more practice. (obligation / requirement)

So 应该 is appropriate for giving advice without sounding overly bossy.

Can 练习 (liànxí) be both a noun and a verb? How do I tell which it is here?

Yes, 练习 can be:

Verb: to practice
- 我们应该多练习。 = We should practice more.
Noun: practice / exercise(s)
- 做练习 = to do exercises
- 一个练习 = one exercise

In 我们应该多做练习, 练习 is a noun because:

It follows 做 (a verb that commonly takes nouns as objects: 做饭, 做作业, 做运动).
The pattern is 做 + [noun] = do + [thing].

So: 做练习 = do exercises, not practice (verb).

If I want to specify a lot of exercises, do I need a measure word with 练习?

When you count or quantify 练习, you normally use a measure word, often 个 (gè) or 道 (dào) (for questions/problems):

一个练习 = one exercise
很多个练习 = many exercises
三道练习题 = three exercise questions/problems

But in 多做练习, you are not explicitly counting; 多 is used adverbially (do more).
If you wanted to emphasize quantity as a noun phrase:

我们应该做很多练习。 = We should do many exercises.
我们应该做更多练习。 = We should do more exercises.

In those, 很多 / 更多 work as modifiers of the noun 练习, and a measure word can be added if you want to be more specific.