De monteur schijnt mijn rem en ketting vanavond nog te kunnen repareren.

Not in this sentence.

The verb schijnen can mean:

• to shine
  
  • De zon schijnt. = The sun is shining.
• to seem / apparently / it is said that
  • De monteur schijnt ... te kunnen repareren.

Here, schijnt means something like:

• seems to
• apparently
• is said to

So the speaker is not stating the repair as a hard fact, but as something they have heard or inferred.

This is a very common Dutch verb chain.

It breaks down like this:

• schijnen = to seem / apparently
• kunnen = to be able to / can
• repareren = to repair

So:

• schijnt te kunnen repareren
  = seems to be able to repair

Each verb adds a layer of meaning:

• repareren = the action
• kunnen repareren = be able to do the action
• schijnt te kunnen repareren = apparently/seemingly be able to do the action

Because schijnen is followed by te + infinitive, and the infinitive here is the whole phrase kunnen repareren.

So Dutch treats kunnen repareren as one verbal unit:

• schijnen te + [kunnen repareren]

And inside that unit, kunnen is a modal verb, and modal verbs are normally followed by a bare infinitive, without te:

• kunnen repareren
• not kunnen te repareren

So:

• schijnt te kunnen repareren = correct
• schijnt kunnen repareren = incorrect in standard Dutch here
• schijnt te kunnen te repareren = incorrect

This is normal Dutch word order.

In a main clause, the finite verb usually comes in second position. Here the finite verb is schijnt:

• De monteur = first element
• schijnt = second-position finite verb

The other verbs, which are infinitives, go toward the end:

• te kunnen repareren

So the sentence structure is roughly:

• Subject + finite verb + middle part + infinitive cluster

That gives:

• De monteur schijnt mijn rem en ketting vanavond nog te kunnen repareren.

This often feels very different from English, where the verbs stay together more easily:

• The mechanic seems to be able to repair...

Here nog adds the idea of still / yet / before the evening is over.

So vanavond nog suggests:

• still tonight
• as early as tonight
• before tonight ends

It often implies that this is sooner than expected, or that there is still enough time.

Compare:

• vanavond = tonight
• vanavond nog = still tonight / already tonight before the night is over

So the sentence suggests the mechanic can apparently repair it tonight already / tonight after all / before tonight is over.

Yes, Dutch allows some flexibility.

For example:

• De monteur schijnt mijn rem en ketting vanavond nog te kunnen repareren.
• De monteur schijnt mijn rem en ketting nog vanavond te kunnen repareren.

Both are possible. The difference is mostly one of emphasis and natural flow, not basic meaning.

In the original sentence, vanavond nog is a nice fixed time expression, and it sounds very natural together.

Because one possessive can apply to both nouns.

So:

• mijn rem en ketting = my brake and chain

Dutch does this just like English:

• my brake and chain
• not necessarily my brake and my chain

You can repeat it for emphasis:

• mijn rem en mijn ketting

But that is not necessary unless you want to be extra clear or contrastive.

Grammatically, rem is just singular brake.

Why singular is used depends on context:

• the speaker may mean one specific brake
• rem can sometimes be used somewhat generally for the braking mechanism
• the situation may simply involve one broken brake

So there is no grammar problem here. It just means brake in the singular.

If the speaker meant more than one brake, they could say:

• mijn remmen en ketting = my brakes and chain

A monteur is a mechanic, technician, or repair person, depending on context.

It is a broad Dutch word used for someone who installs, maintains, or repairs things such as:

• bicycles
• cars
• machines
• appliances
• technical equipment

In this sentence, because of rem and ketting, many learners would picture a bike mechanic, though the sentence itself does not force that 100%.

Yes, at least a little.

Using schijnt shows that the speaker is not fully committing to the statement as direct fact. It often means:

• that’s what I hear
• apparently
• it seems

So compare:

• De monteur kan mijn rem en ketting vanavond nog repareren.
  = The mechanic can repair my brake and chain tonight.
  More direct.

• De monteur schijnt mijn rem en ketting vanavond nog te kunnen repareren.
  = The mechanic apparently/seemingly can repair my brake and chain tonight.
  More indirect or tentative.

Yes, they are related in meaning, but not always identical.

• lijken = to seem / appear
• schijnen = to seem / apparently / be said to

In many sentences they are close:

• De monteur lijkt mijn rem en ketting vanavond nog te kunnen repareren.
• De monteur schijnt mijn rem en ketting vanavond nog te kunnen repareren.

But schijnen often has a stronger sense of according to what people say or apparently, while lijken more often describes the speaker’s own impression.

That is not a hard rule in every situation, but it is a useful tendency to remember.

Yes, that is very useful.

A good pattern is:

• schijnen te + infinitive
• and if that infinitive is a modal verb:
  • schijnen te kunnen + main verb
  • schijnen te moeten + main verb
  • schijnen te willen + main verb

Examples:

• Hij schijnt te kunnen zwemmen.
  = He apparently can swim.
• Ze schijnt te moeten werken.
  = She apparently has to work.
• Hij schijnt te willen vertrekken.
  = He apparently wants to leave.

So in your sentence, schijnt ... te kunnen repareren is a very standard Dutch structure.