These seem to be used in the same contexts a lot, but there are some cases where their usage differs.
試す - Trying something with the motive of discovering something's qualities.
試みる - Trying something without knowing what the result will be.
Is seen used in more formal situations too.
So for these sentences, based on the goal it would change. 試す would show ...
お辞儀: a bowing gesture
敬礼: a saluting gesture (this)
礼儀: courtesy; manner
礼: 1) =お辞儀; 2) =礼儀; 3) word of thanks; 4) token of gratitude, favor given in return
会釈: a casual, shallow and silent bow
一礼: a suru-verb, "to bow (once, casually)"; rarely used as a regular noun
(船首: the bow of a ship (this))
お辞儀 is a plain term that refers to ...
As Leebo has mentioned, this is a relative clause. In other words,
pretty much mean the same thing. This is a good explanation of が and の usage.
Simply, 思い込む has several meanings:
[common] to firmly (mis)believe; to to make a (wrong) assumption
[uncommon] (≒決心する) to resolve; to make up one's mind
[rare/obsolete] (≒思い詰める) to think hard; to worry deeply; to be obsessed
大辞林 says 思い込み can mean "firm resolution", but at ...
Since you haven't provided any context, let me point out that ある could be used in several different ways, like
stating that something exists (the literal meaning)
stating that something is reasonable/common/good (cf. それはないよ).
あるちゃあるけど is sometimes used in a longer version
with a loose translation being
You can't say that it's not a thing,...
Believe you're missing a letter:
It's an extremely informal way of saying:
I won't deny that we have it
But the underlying connotation is that item is not available for sale/use, or it's reserved for some reason.
You could consider two interpretations.
Omission from double nominative このカメラは 対象が よく写る
よく写る itself expresses a capability.
Incidentally, you can't really use 写す when the camera is the subject because it's an inanimate object, which doesn't take actions at its own will.
I rather read your remark on English connotation interestingly, but anyway, there do exist some cases where the superficial form and the meaning 違う, in any language.
Effectively, you can just take 違う as Japanese way to say "it's not", because:
あなたはプログラミングしますか？ [ordinary verb]
→ いいえ、プログラマーじゃありません。 (too long!)
I was watching an anime in which at some point a letter is shown.
The situation is like the brother of the reader of this letter is hiding a card in a shopping mall and the reader must find it. A card is obviously a thing and not a living being.
So, it seems that 捜 is used in a "hunt" way, searching things or people that you don't know where they are but ...
Your first example is special. 言うこと is almost an idiom (I'm sure it is except I don't find it in dictionaries) that should be understood as "(one's) words" rather than what it looks like. Thus 弟の言うことを聞いた is "I listened to my brother's words", or in English, even "I listened to my brother". If you said 弟の言ったことを聞いたばかりに～, it'd mean ...
In English, "most" works both as an adjective (i.e, a noun modifier) and as an adverb (i.e., a verb/adjective modifier). Compare the following sentences:
He has the most money.
He is the most honest person.
He ran the most.
You can see "most" is adjectivally modifying a noun ("money") in 1, adverbially modifying an adjective ("honest") in 2, and ...