ملقط النفايات قديم. مع ذلك يستعمله المتطوع.

This is an iḍāfa construction, often called a construct phrase.

• ملقط = picker / grabber / tongs
• النفايات = the trash / waste

Together, ملقط النفايات literally means the picker of the trash, i.e. the trash picker / litter grabber.

In an iḍāfa:

• the first noun usually does not take ال
• the second noun shows whether the whole phrase is definite or indefinite

So because النفايات is definite, the whole phrase ملقط النفايات is definite.

Yes. Arabic often puts two nouns together in an iḍāfa to express relationships that English can express in several ways:

• trash picker
• picker of trash
• garbage grabber

So ملقط النفايات is a noun + noun structure where the second noun specifies the first.

This is very common in Arabic, for example:

• باب البيت = the door of the house / the house door
• كتاب الطالب = the student’s book

Because قديم agrees with the head noun, which is ملقط, not with النفايات.

The head noun is:

• ملقط = masculine singular

So the adjective or predicate is also masculine singular:

• قديم = old

Even though النفايات is plural, it is only the second part of the iḍāfa and does not control the agreement here.

The same thing happens with the pronoun in يستعمله: the ـه also refers to ملقط, not to النفايات.

In Arabic, the present-tense copula is usually omitted.

So:

• ملقط النفايات قديم = The trash picker is old

Arabic does not normally insert a separate word for is in this kind of present-tense sentence.

But in other tenses, Arabic does use a verb, for example:

• كان ملقط النفايات قديمًا = The trash picker was old

So the first sentence is a normal Arabic nominal sentence.

Because قديم here is the predicate of the sentence, not an adjective inside the noun phrase.

Compare:

• ملقط النفايات قديم = The trash picker is old
  
  • here قديم is the predicate
• ملقط النفايات القديم = the old trash picker
  
  • here القديم is an adjective modifying the noun phrase

So:

• without ال: old as the statement
• with ال: the old as part of the noun phrase

That is an important difference in Arabic.

مع ذلك is a common expression meaning:

• however
• nevertheless
• even so
• despite that

Literally, it is something like with that, but you should learn it as a fixed expression.

So:

• ملقط النفايات قديم. مع ذلك يستعمله المتطوع.
  = The trash picker is old. Nevertheless, the volunteer uses it.

يستعمله has two parts:

• يستعمل = he uses / he is using
• ـه = him / it

So يستعمله means he uses it.

In this sentence, the ـه refers to ملقط.

This kind of attached object pronoun is very common in Arabic:

• أراه = I see him / it
• أعرفه = I know him / it
• أستعمله = I use it

Because Arabic commonly uses attached object pronouns.

So instead of saying something like uses it with two separate words, Arabic often says it as one unit:

• يستعمله = uses-it

This is completely normal Arabic grammar.

A separate object pronoun can exist in some contexts, but attached pronouns are much more basic and common in sentences like this.

Because the pronoun refers back to ملقط, not to النفايات.

• ملقط = masculine singular
• therefore ـه = masculine singular it

So the logic is:

• The trash picker is old. Nevertheless, the volunteer uses it.

The it is the picker, not the trash.

This is a very common place where learners need to remember that the first noun in an iḍāfa is the main noun.

Because Arabic often uses verb–subject order.

So:

• يستعمله المتطوع = literally uses it the volunteer

But in natural English, that becomes:

• The volunteer uses it

Arabic can also say:

• المتطوع يستعمله

That order is also possible, but it gives a slightly different feel, often with more focus on the volunteer.

So both are possible, but verb first is very common in Arabic.

In fully inflected formal Arabic, yes, there would be case endings, for example:

• ملقطُ النفاياتِ قديمٌ
• مع ذلك يستعملُه المتطوعُ

But in normal Arabic writing, these short vowels are usually not written.

That means learners have to recognize the grammar from the structure, not from visible endings alone.

Very roughly:

• ملقطُ = subject
• النفاياتِ = second noun in iḍāfa, therefore genitive
• قديمٌ = predicate
• يستعملُه = verb + object pronoun
• المتطوعُ = subject of the verb

So the unvowelled spelling is normal, even though the grammatical endings are still understood.