Uh, no, this いう doesn't mean "to say." You can also say 一緒に勉強するのはどう？ in this case. っていう/という has a function to make it sound like "the clause in parentheses"; 「一緒に勉強する」っていうのはどう？
ってのは is just the contraction/colloquial form of というのは/っていうのは, or a Tokyo dialect if pronounced as ってぇのは. It's rarely used by female, I suppose. ～してはどうですか？ isn't basically used for "Why don't we/How about we," and could sound somewhat arrogant.
ってのは is just the contraction/colloquial form of というのは/っていうのは, or a Tokyo dialect if pronounced as ってぇのは. It's rarely used by female, I suppose. ～してはどうですか？ isn't basically used for "Why don't we/How about we," and could sound somewhat arrogant.
I was wondering about that. 「なんで～しないの？」ていう感じで失礼にとられそうな気がしますね。 しては :I know there is similar grammar, it just sounds a little unexpected not being nominalized before the は.
Tokyo dialect if pronounced as ってぇのは. That's probably were it was picked up, ignoring the "natural" emphasis through long consonants we don't have in English.