Austin is right. '-고는 하다' is used to express when same thing repeats over and over or usual habits. So it should be '배우고는 했어요'. As Austin mentioned, native Korean people commonly shorten words for easier pronunciation, it is commonly used as '-곤 하다' in speech. However, you cannot use it in formal setting such as writing. Here are more sample sentences using '-고는 하다'. e.g. 우리는 매일 산책을 하고는 했어요. ( We used to walk every day.) 저는 여름마다 한국에 가고는 했어요. (I used to go to Korea every summer.)

That's not a grammar point I'm familiar with. It might be a typo of ~곤 하다. That's what would make the most sense to me in these contexts. It could also be spoken Korean and the person just dropped the ㄴ to make it easer to say? Native speakers of any language often shorten words or say things "incorrectly" just for convenience. I'm not a native speaker, so perhaps a native Korean speakers will have a better answer :)