1. The speaker is explaining why he thinks that students who go to study abroad are likely to be only children (i.e. have no brothers or sisters).
だって親の方も結構大変じゃないですか。子供を海外留学させろのって、いろんな面で。
I understand fragments of this, but I cannot get the overall sense. What does it mean?
Because it is tough on the parents' side, too, to let their children go to study abroad, in many ways.
2. うちの父なんか、子供がみんな一人立ちしたからほっとしたなんて言ってますよ。今年61になったんだけど、老後の生活に備えて料理でも習おうかななんて言っちゃって。
My translation: "My father says he's relieved now that all his children are living independent lives. He was 61 this year, and he's even talking about learning how to cook so he can get by in old age."
a) At the end, is 習おうかななんて = 習おう + か + な + なんて?
習おう + かな + なんて
b) At the start, does なんか have much actual meaning, or is it just a filler word?
It is suggest to mention it as an example.
テ」ツ?ェテ」窶壺?愿」ツ≫?ケテッツシツサテ・窶ーツッテ・ナ?ツゥテッツシツステ」ツ?ョテヲ窶楪湘・窶伉ウ - テ・窶コツスティツェナセティツセナセテヲ窶コツク - gooティツセナセテヲ窶コツク
なんか[副助]
《代名詞「なに」に副助詞「か」の付いた「なにか」の音変化から》名詞、名詞に準じる語、活用語の連用形、一部の助詞などに付く。
1 一例を挙げて示す。…など。「この着物―お似合いです」「映画―よく行く」
c) Similarly for なんて, does it have a definite meaning here of "(he says) things such as...", or is it more like just a filler?
Quoting with some feelings. In this case, I assume, the speaker thinks what his father says is rather expected.
テ」ツ?ェテ」窶壺?愿」ツ?ヲテッツシツサテ・窶ーツッテ・ナ?ツゥテッツシツステ」ツ?ョテヲ窶楪湘・窶伉ウ - テ・窶コツスティツェナセティツセナセテヲ窶コツク - gooティツセナセテヲ窶コツク
3. The speaker has been asked whether his parents supported his plans to study overseas.
いや~、どうかなあ。うちの父なんか結構頑固で、言い出すとなかなか聞いてくれないんですけどね。でも留学の方は、どういうわけか、すんなりオッケーしてくれましたね。
My translation: "???. My father is very stubborn, and he's not very ready to listen to suggestions. But where studying abroad was concerned, for some reason he said OK straight away."
a) Apart from the missing part, is this OK?
Yes.
b) What does いや~、どうかなあ mean? I thought it had a negative sense, like "no, not really", but this doesn't seem to fit with what follows.
It means "well, I wonder".
c) Is 方 = ほう?
Yes
d) Same question as before about なんか. Is it more or less just a filler word?
Same answer to another question.
4. (This is a different person speaking now)
父親は頑固だし、母親は行かせてやってもいいと思ったみたいだけど、父の言うことにあんまり反対しないから。
My translation: "My father's stubborn; my mother seemed to think it was OK to let me go [overseas to study], but she never opposes what my father says."
a) Is 母親 the subject of 思った?
Yes.
b) Does みたい mean "seemed" here?
Yes.