Hi eeky,
1. 新米のくせに大きな顔をするな。
I understand that 大きな顔をする means roughly "don't act like you are so important", but what does 新米のくせに mean (literally and/or idiomatically)?
~くせに is to express one's negative feeling, often accusation, looking down upon, etc.
"Don't act like you are so important, you're nothing but a newbie", for example.
縺上○縺ォ縲千剿縺ォ縲托シサ謗・蜉ゥ・ス縺ョ諢丞袖 - 蝗ス隱櫁セ樊嶌 - goo霎樊嶌
くせに【癖に】
[接助]《名詞「くせ(癖)」+格助詞「に」から》活用語の連体形および格助詞「の」に付く。
非難や不満の気持ちを込めて逆接条件を表す。
にもかかわらず。のに。くせして。「弱い―強がりを言う」「本当は好きな―、そぶりも見せない 」
3. 意志疎通のはかり方は、文化によって決まるからである。
I believe this is saying something about methods of mutual understanding being determined by cultural considerations. However,
a) Assuming that はかり is from はかる, in what sense is this verb used here?
~を図る means to make efforts to do something, or to commit something.
縺ッ縺九k縲占ィ医k・城㍼繧具シ乗クャ繧具シ丞峙繧具シ剰ャ 繧具シ剰ォョ繧九代・諢丞袖 - 蝗ス隱櫁セ樊嶌 - goo霎樊嶌
意思疎通を図る means to try to communicate, or simply to communicate.
b) I don't know how, but I seem to have confused myself about the intent of からである here.
Does it just make the sentence into an explanatory one (like prefixing the sentence with "the reason is..." in English)?
Yes. And there must be some 'conclusion' before this sentence.
4. Someone is introducing themselves and has said that they come from 安城市.
「安城市ってどこ?」ってよく聞かれるんですけど、「新幹線の駅があるよね」と言ってくれる人 もいます。
My translation: "It's often asked 'Where is Anjō', and some people say 'Well, it has a Shinkansen station!'."
Here I am unsure who or what is the subject of 聞かれる (which I'm assuming is passive). Is it 「安城市ってどこ?」 (i.e. "「安城市ってどこ?」 is often asked"?) Is it the speaker (i.e. "I am often asked...")? Is it people in general? At first I thought it was the speaker, but then I hit 人 and I couldn't connect "I am often asked" with "there are people who say...".
聞かれる is passive, the speaker is often asked 'where is 安城市?'.
That means most people do not know, have no idea where 安城市, which might be disappointment for the speaker - assuming it's his home town, and even though he knows that it's a small, minor city yet it is a bit sad for him that people do not know about it.
Then there are some people who know 安城市, saying "it has a Shinkansen station, doesn't it?"
It's something he is pleased to hear, thus 言って「くれる」 - he feels as if he received some favour.
Hope the above helps.