というのに means "(al)though (that)".
While she hadn't eaten in days, she looked healthy.
Word Note: Remember that 食う is the vulgar form of 食べる.
While deserts receive almost no rain, it looks like a few plants are somehow able to manage to live there.
～ということは means "that means" and ～というのは means "the reason that". Both are placed at the beginning of a sentence. In slang ～ということは may contract to ～ってことは. In extreme slang the entire phrase can be shortened to ～つうか.
That means he probably hasn't returned home from his business trip yet.
The reason for that is that he has been deceived by his friends before.
The reason for that was that he started a new business.
～とはいえ/いいながら/いうものの means "even though". The exact meanings are slightly different. The first emphasizes contrast with what "can be said", the second is like "while though", and the last is like "although...say". They can more or less be translated as "even though" in English.
Even though one may call it the rainy season, there are still also clear days.
Culture Note: The 梅雨 lasts from June to July. ばいう is a 書き言葉 reading.
Though (this) may be delicious, there are bad things included as well.
Even though she is young, she is quite experienced.