ない、-て

So I've encountered a couple of sentences I don't quite understand while practicing with iknow.

1. 私たちは駅の階段をかけ上がった。
We ran up the station steps.

What is the kake here?

2. この部分は問題ないです。
There's no problem with this part.

What is the nai here? And how does it compare to if it and desu were to be replaces with janaidesu or dehaariamsen?

3. 生まれて初めて布団を敷いた。
I laid out the futon for the first time in my life.

What does the -teform in umarete mean here? Is umaretehajimete a set phrase perhaps?

4. 良い友達がいて私は幸せだ。
I'm happy that I have good friends.

それを聞いて安心しました。
I was relieved to hear that.

The -teform in these sentences are they causative -teform? (I think I've read that something is called causative te-form somewhere, perhaps here.)
How would it compare to if they were to be replace with dictionary form (iru and kiku) and then append kara or node? E.g. the second one would be それを聞くから安心しました.

1. Run. Compound verb formed from kakeru and agaru.

2. Plain form of arimasen, with desu added for politeness

3 and 4. The continuative tense. Used to tack verbs together sort of like "and".

2)
問題ない is 問題がない/ありません, whereas 問題じゃない/ではありません means "not the problem/issue".

3)
生まれて modifies 初めて. You can think it as 生まれてから初めて "first time since I was born".

4)
The -te form indicates the reason/cause of the main clause in this case, so it can be replaced with から/ので. However, you need to pay attention to the tense.

良い友達がいるから/ので、私は幸せだ。
それを聞いたので/から安心しました。

This usage is different from #3. See the examples below.

赤ちゃんが、生まれて初めて歩いた。
A baby stood up and walked for the first time in his/her life.

子供が生まれて、初めて親の気持ちがわかった。（＝子供が生まれたから/ので、初めて親の気持ちがわかった。）
Because/Since I have a child, I finally understand my parents' feelings.

Thank you both, very much

1. Had no idea about this verb and could not find it.

2. Oh, of course it's arimasen. Just could not see it. @Toritoribe, can the two different sentences then sort of be, This part does not have a problem(direct translation not very natural), and this part is not the problem. Translating aru as have and desu as is.

3. Hmm, I have to remember that -te can modify words. I think you have told me this before toritoribe. I have only read about "short form" verbs modifying, not with the -te form.

4. Yea, the tense part was just my screw up...
Your examples are great. Without the comma in the second example could it be translated, perhaps strangely, to The child for the first time in his/her life understood a parents love?

Also, this reason/cause usage of the -te form is that what is being used in this sentence:
バスに乗って、会社に行きます。
Or is this a different usage of the -te form?

2)
In the example sentence, この部分 is not the subject but the "location" where the problem is. この部分は is the topicalization of この部分に.

問題がある。
There is a problem.

問題（が）ない。
There is no problem.

この部分は/に/には問題がある。
There is a problem with this part.

この部分は/に/には問題（が）ない。
There is no problem with this part.

However, it's not so simple about 問題だ/問題ではない because 問題だ can mean "to have a problem".

この部分は問題だ。
This part has a problem.

この部分が問題だ。
This part is the problem.

この部分は問題ではない。
This part can't be a problem. = We don't need to warry about this part.
(This sentence implies "another part has a problem.")

この部分が問題ではない。
This part is not the problem (that we are talking about).

3)
You can think "～して･･･" as "･･･ after/since ~".

買って二日目に財布を落とした。
I dropped my wallet on the second day after buying it.

日本に来てすぐ彼女に会った。
I met her soon after coming to Japan.

staren said:

Without the comma in the second example could it be translated, perhaps strangely, to The child for the first time in his/her life understood a parents love?

Click to expand...

Exactly.

staren said:

Also, this reason/cause usage of the -te form is that what is being used in this sentence:
バスに乗って、会社に行きます。
Or is this a different usage of the -te form?

Click to expand...

That's not the reason/cause but the means. It can be replaced with で.

船に乗ってアメリカに行った。 ＝ 船でアメリカに行った。

そろばんを使って計算した。 ＝ そろばんで計算した。
I calculated using an abacus.

2. I wouldn't say I completely understand it but more reading will probably help me understand it. Just need some time and more exposure.

3,4. Great. I thought it was like that. As always, thanks, the examples are great!