Literally : 'say X but' e.g. "Say it's tough, but because it only lasts three months ..."
In the first example, the highlights are positioned wrong. 「大変だといっても3ヶ月間だけですから。」 (It's tough, but it only lasts three months.) The "but" should be highlighted. Also, the given English translation adds too many extras, such as "you can say" and "keep in mind." Those are not present in the Japanese sentence. Also, can we try to remember periods?
Example #4409 is in the wrong grammar entry. It should be in たとえ〜ても, which means "even if." といっても by itself means "yet, however, but." Also, the correct translation is, "Even if I say that there's a convention, I really shouldn't ask them for so much money."
Example #4412, it should be "I was wrong after all." Example #5995 doesn't have an English translation, which should be "Although there's a garden, it's rather tiny."
As shown in Example #3111, another version of 〜と言っても is 〜と言ったって. This is no less formal and has exactly the same meaning. A less formal and less common version is 〜ったって, which is placed after a plain affirmative verb, eg 帰るったって.
Shouldn't the grammar formation for なＡ・Ｎ be: なＡ・Ｎ + だ＋といっても as used in #277 and #5949? Or perhaps なＡ・Ｎ + (だ)といっても ?
isn`t this the Vても form meaning even if?
彼はいい人だっても、好きにならないよ even if he was a good person, I wouldn`t like him.
この上着が７５％割引になっても、買えない even if that jacket was 75% off I couldn`t buy it.
「〜」と言っても ＝ even if you say”〜”
The English translation for #6725 looks way off. Should probably be something like, "I don't really like sweet things. But then again, that doesn't mean I never eat cookies or cake.
Ok Coneさん, I changed the English translation close to your version. Thanks for contributing.
A:It's good that you can visit many places, isn't it? B:I travel but for work, so I can't have fun.
The translation for ex #5995 was inaccurate. 額（ひたい）in this context means "forehead", not "amount". So you get, "the forehead of a cat", an amusing Japanese way of saying "tiny". "The amount of a cat" makes absolutely no sense. Think harder next time about your translations.
I think the Header Meaning should say more than just "but"...the idea I see most often with this is:
A is X, but it is only a little bit X
In other words, the second part serves to deprecate the first part ("We have a garden, but it is only a tiny garden." or "I travel a lot, but only for work ( so it isn't much fun).")
Although I said A but not-so-important B actually happened is correct translation of it.