Sofie moet de tas met boeken alleen tillen.

This is normal Dutch word order.

In a main clause, the finite verb usually goes in second position. Here, moet is the finite verb.

When Dutch uses a modal verb like moeten with another verb, the second verb stays as an infinitive and usually goes to the end of the clause.

So:

• Sofie tilt de tas. = Sofie lifts the bag.
• Sofie moet de tas tillen. = Sofie must lift the bag.

That is why you get Sofie moet ... tillen.

Moet is the present tense singular form of moeten.

In this sentence, it means something like:

• must
• has to
• needs to

So Sofie moet ... tillen means that Sofie is required or expected to do the lifting.

A few forms of moeten are:

• ik moet = I must / I have to
• jij moet = you must / you have to
• hij/zij moet = he/she must / has to
• wij moeten = we must / have to

Because after a modal verb such as moeten, Dutch uses the bare infinitive.

So you say:

• Sofie moet tillen
• not Sofie moet te tillen

This is similar to English:

• She must lift
• not She must to lift

By contrast, some other Dutch verbs do require te:

• Sofie probeert de tas te tillen. = Sofie tries to lift the bag.

Because tas is a de-word in Dutch.

Dutch nouns are either:

• de-words
• het-words

You simply have to learn which article goes with each noun. For tas the correct article is de:

• de tas = the bag

A useful thing to remember: all plural nouns take de, but in the singular you have to learn whether a noun is de or het.

Here met boeken describes de tas.

So de tas met boeken means:

• the bag with books
• the bag containing books

In other words, met boeken belongs with tas, not with tillen.

English does the same thing:

• the bag with books

Because boeken here is being used in a general, indefinite way.

• met boeken = with books / containing books
• met de boeken = with the books

So:

• de tas met boeken = the bag with books
• de tas met de boeken would sound more like the bag with the books, referring to specific books already known from the context.

Here alleen most naturally means alone or by herself.

So the idea is that Sofie has to lift the bag without help.

However, alleen can also mean only, depending on context and word order. That is one reason learners often ask about it.

In this sentence, the most likely meaning is:

• Sofie must lift the bag with books by herself.

Yes, it can.

Compare these:

• Sofie moet de tas met boeken alleen tillen.
  Most naturally: Sofie must lift the bag with books alone / by herself.

• Sofie moet alleen de tas met boeken tillen.
  This can more easily suggest: Sofie must lift only the bag with books.

Dutch word order often helps show what an adverb like alleen is focusing on.

If you want to make by herself extra clear, Dutch speakers may also say:

• Sofie moet de tas met boeken zelf tillen.

Tillen usually means to lift or to pick up, often with the idea that something is heavy.

Dragen usually means to carry or to wear.

So:

• tillen = lift, raise
• dragen = carry

In this sentence, tillen suggests the action of physically lifting the bag, possibly because it is heavy. Since the bag has books in it, that makes sense.

Yes, that is basically what alleen suggests here: she does it on her own, without assistance.

Dutch does not always sharply separate:

• alone
• by herself
• without help

The exact nuance comes from context. In a sentence like this, the normal interpretation is that Sofie has to handle the lifting herself.

Sofie is the subject, but not simply because it comes first. It is the subject because it is the person doing or being required to do the action.

Here:

• Sofie = subject
• moet = finite verb
• de tas met boeken = object
• alleen = adverb
• tillen = infinitive

Dutch often puts the subject first in a normal statement, but other elements can come first too. The subject does not always have to be the first word.