Fulloni linum album traditur, ut id lavet et mollius reddat.

Because Fulloni is dative singular of fullo, fullonis (fuller, a person who cleans and treats cloth).

Here it is the indirect object of traditur:

• linum = the linen
• traditur = is handed over
• fulloni = to the fuller

So the basic idea is: The white linen is handed over to the fuller.

Traditur is the present passive of trado, tradere (hand over, deliver).

So:

• tradit = he/she hands over
• traditur = it is handed over

Since linum album is the thing being delivered, it becomes the subject of the passive verb:

• Linum album traditur = The white linen is handed over

Latin often uses the passive when the action matters more than the person doing it.

Linum album is nominative singular neuter, because it is the subject of traditur.

Both words match:

• linum = neuter singular
• album = neuter singular adjective agreeing with linum

So album means white, and it agrees with linum in gender, number, and case.

Yes. Latin word order is much freer than English word order because the endings show the grammar.

So linum album and album linum would both mean white linen.
The usual reason for one order or another is style, emphasis, or rhythm, not a change in basic meaning.

In this sentence, linum album is a perfectly normal order.

Here ut means so that or in order that.

It introduces a purpose clause:

• ut id lavet et mollius reddat
  = so that he may wash it and make it softer

So the fuller receives the linen for a purpose.

Because after ut in a purpose clause, Latin normally uses the subjunctive.

So:

• lavet = may wash
• reddat = may make / render

This does not necessarily mean doubt. It is just the normal grammar for a purpose clause:

• ut ... lavet et reddat = so that ... he may wash and make softer

The subject is understood to be the fuller.

Latin often leaves subject pronouns unstated when they are clear from context. Here the sense is:

• Fulloni linum album traditur, ut is id lavet et mollius reddat.

Latin does not need to say is (he) explicitly, because the context already tells us who is meant.

So the meaning is:

• The white linen is handed over to the fuller, so that he may wash it and make it softer.

Id is the accusative singular neuter form of is, ea, id (he, she, it / this, that depending on context).

It refers back to linum, which is neuter singular. So:

• linum = neuter singular
• id = it (matching linum)

That is why Latin uses id, not eum:

• eum would be masculine
• id is neuter

Latin often can omit an object if it is obvious, but here id makes the structure clearer:

• ut id lavet et mollius reddat

It serves as the object for both verbs:

• wash it
• make it softer

So id helps mark clearly what is being washed and softened.

Mollius is the neuter singular comparative form of mollis (soft).

Here it means softer.

It goes with the object understood from id:

• id ... mollius reddat
• make it softer

So mollius is not just a random adverb here. It is best understood as a predicate adjective/complement with id after reddat:

• reddere aliquid mollius = to make something softer

In form, mollius could sometimes look like an adverbial comparative, but here the meaning and construction point to softer, not more softly.

Why?

Because reddere often takes:

• an object
• plus a word describing the new state of that object

So:

• id mollius reddat = make it softer

If it meant do the action more softly, the sentence would be expressing manner, but that is not what reddere normally does here.

They are two coordinated verbs joined by et:

• lavet = wash
• et = and
• reddat = make

Both depend on ut, so both are part of the same purpose clause:

• so that he may wash it and make it softer

The same subject (the fuller) and the same object (id, it) apply to both verbs.

Yes. The main verb traditur is present passive, and the purpose-clause verbs lavet and reddat are present subjunctive.

This gives a general or vivid present sense:

• The white linen is handed over to the fuller, so that he may wash it and make it softer.

It can sound like a general statement about what is done with linen, not just a single one-time event.

It can mean either, depending on context. Here, because it is being handed to a fuller to be washed and softened, it clearly means linen cloth or linen fabric, not the flax plant itself.

So context tells you which English word is best.

Grammatically, album is simply an adjective modifying linum:

• linum = linen
• album = white

So it matters because it must agree with linum. But beyond grammar, it also adds meaning: it is specifically white linen that is being delivered.

Yes, that is often the best way to read Latin.

Start with the main clause:

• Linum album traditur fulloni
  = The white linen is handed over to the fuller

Then add the purpose clause:

• ut id lavet et mollius reddat
  = so that he may wash it and make it softer

Putting it all together:

• The white linen is handed over to the fuller so that he may wash it and make it softer.

That step-by-step approach is very useful for Latin reading.