Ok, let's try to fix this:
The problem with some uses of "only" is that is is subjective, it is used to show that you
think something lis less than it should be, and others may agree with you or not, or have a different opinion.
For this reason, there are actually many grammatically correct answers to your practise sentences, it just depends on what you are trying to say.
Also, you seem to believe that each problem can be solved by inserting "only" somewhere in the sentence, but this seems to not be the case.
- John only had three pencils and lost two.(John)
-Your first answer was correct also, it just depends on what you want to say. You could say: Mary lost every single pencil she had; John had three pencils, and only lost two.
But the first answer is better, when something decreases bya majority, we should not say "only".
He had 200 dollars and only lost 150. (wrong)
He had ten books and only sold four. (right)
- No other girl but Ellen only was late for school.
-"Only" can mean "no other" I think the best answer here would be: Only Ellen (and no other girl) was late for school.
- Sam was the youngest in his class, as he was only seven and all the others were eight or more years old.
-This is correct, "only" here means we feel the number (age) is not enough, smaller/less in comparison. - It rained on Wednesday only . Every other day in the week was fine.
-I disagree with the other poster. Without the second sentence "It only rained on Wednesday" could mean "it only rained, and did not thunder" but the second sentence is there to clarify.
Only belongs before rained, meaning it rained on Wednesday (and no other day). In English the adverb can sometimes be placed before or after the verb, we could say "It rained only on Wednesday." but this makes our sentence convoluted, and less clear.
In reality, we would never use this sentence alone, so it is clear that "It only rained on Wednesday" means "and not on any other day".
- If you only asked me I could have gone with you.
-This is a colloquial use of "only" and does not really have any of the original meanings (this alone, and not that, a number that is smaller than expected, or too small). Here it means it would have been better or simpler to do the action, it is always placed immediately before the verb or verb phrase, and can be replaced (colloquially) with "just".
i.e.: "If you just asked me, I could have gone with you."
- There had been two hundred thousand soldiers in the army. But only half of them now remained after the battle.
Here I have to wonder which book you are using, there are a few things wrong with this sentence, mainly it uses a conjunction (but) to begin the second sentence and it confuses verb tenses (they now remained/ now they remained). This may be some oddity of British English, perhaps.
Again, your placement of only seems correct. I would completely restructure the sentence as follows:
There had been two hundred thousand soldiers in the army, but now(,) after the battle, only half of them remain.
-OR-
There had been two hundred thousand soldiers in the army, but after the battle, only half of them remained.
I moved the phrase of time because it's clear the sentence is trying to emphasise that things are different after the battle: but now, after the battle...
and not "they remained, after the battle" we are not emphasising that they remained.