Hún var til í að bíða á meðan ég pumpaði dekkið og athugaði keðjuna.

Vera til í að is an idiomatic expression meaning to be willing to, to be up for, or to be okay with doing something.

So:

• Hún var til í að bíða = She was willing to wait / She was okay waiting

You should learn til í here as part of the whole expression, not word by word. The literal meanings of til and í do not help much on their own in this phrase.

Examples:

• Ég er til í kaffi. = I’m up for coffee.
• Ertu til í að koma? = Are you willing / up for coming?

Var is the past tense of vera (to be).

• hún er = she is
• hún var = she was

Since the whole sentence is talking about a past situation, Icelandic uses the past tense:

• Hún var til í að bíða... = She was willing to wait...

If it were happening الآن / in the present, you would say:

• Hún er til í að bíða... = She is willing to wait...

Here að is the infinitive marker, like English to before a verb.

• að bíða = to wait

So:

• Hún var til í að bíða = She was willing to wait

This is very common in Icelandic after expressions that are followed by an infinitive:

• langa að = want to
• reyna að = try to
• vera til í að = be willing to

Do not confuse this að with the preposition að meaning at/to. Here it is simply marking the infinitive.

Á meðan means while.

In this sentence:

• á meðan ég pumpaði dekkið og athugaði keðjuna = while I pumped the tire and checked the chain

It introduces a time clause: one thing happened during the time that another thing was happening.

You will also see meðan by itself in some contexts. In everyday Icelandic, á meðan is very common for while / in the meantime.

Both are 1st person singular past tense verbs, because the subject is ég (I) and the actions happened in the past.

• ég pumpaði = I pumped
• ég athugaði = I checked

So:

• ég pumpaði dekkið = I pumped the tire
• ég athugaði keðjuna = I checked the chain

The ending -ði is very common in the past tense of many weak verbs in Icelandic.

Because Icelandic is using the definite form: the tire and the chain, not just a tire and a chain.

• dekk = tire

• dekkið = the tire

• keðja = chain
• keðjan = the chain
• keðjuna = the chain (accusative form)

Icelandic often adds the equivalent of the as an ending on the noun instead of using a separate word.

Because the nouns belong to different grammatical genders and declension patterns.

• dekk is neuter
• keðja is feminine

Also, both nouns are direct objects here, so they appear in the accusative case.

That gives:

• dekkið = the tire (neuter accusative singular; same form as nominative here)
• keðjuna = the chain (feminine accusative singular)

So the different endings are normal Icelandic case-and-gender behavior.

Because it is the direct object of athugaði (checked).

In Icelandic, many verbs take a direct object in the accusative case. Here:

• athuga eitthvað = check something

So:

• athugaði keðjuna = checked the chain

Likewise:

• pumpaði dekkið = pumped the tire

Both dekkið and keðjuna are objects of verbs, so accusative makes sense here.

Yes. Pumpa is a normal verb in modern Icelandic, although it is a loanword in origin.

• að pumpa = to pump
• ég pumpaði = I pumped

In everyday speech, especially about bike tires, this is completely natural. Depending on context, you may also hear a fuller expression like pumpa í dekkið or pumpa upp dekkið, but pumpaði dekkið is understandable and natural in casual usage.

It breaks down like this:

• Hún var til í að bíða = She was willing to wait

• á meðan = while

• ég pumpaði dekkið = I pumped the tire

• og athugaði keðjuna = and checked the chain

So the full structure is:

[main clause] + [while-clause]

More literally:

• She was willing to wait while I pumped the tire and checked the chain.

Yes, depending on context, that can be a very natural translation.

Hún var til í að bíða can suggest:

• she was willing to wait
• she didn’t mind waiting
• she agreed to wait
• she was up for waiting

The exact English wording depends on tone and context. The Icelandic expression is slightly flexible, and often sounds casual and cooperative.

Because both verbs have the same subject: ég (I).

So Icelandic, like English, does not need to repeat the subject:

• ég pumpaði dekkið og athugaði keðjuna = I pumped the tire and checked the chain

You could repeat ég, but it would usually be unnecessary here. The second verb is simply coordinated with og (and) and shares the same subject.