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well done for using it, and I up-voted you for that. In English English I've only heard it used ceremoniously or for comedy effect such as inflecting irony.
you're probably right however I chose to answer his exact question as its not in my opinion correct to second-guess someone's possible meaning and then answer that. I don't recall seeing old text with "badly" in it, so possibly they never co-existed in the way that he is thinking of . As far as I know, the archaic form of bad is "ill" as in "ill-informed", "ill-done" more as an analogue to "well-"
Following @livrecache above - i think this is archaic in that it refers to "name" in the sense of "reputation". As he/she says there is a phrase "making a name for yourself", meaning becoming well-known and with the implication of it being a favourable reputation. I therefore suspect that "having something of a name" means "mildly famous/well-respected for doing <whatever>"
In Engiish english, i think "woman" wounds quite rude when communicating with someone. If I'm with my kids and we are paying in a shop, I always say that "we have to pay the lady before you can open it".