Usually 'the' would refer to a particular chemist, etc., but sometimes is used in instructions to mean whichever chemist is applicable e.g. 'take this prescription to the chemist and get the medicine' - it could be any chemist. It is equally correct to say ''take this prescription to a chemist and get the medicine".

AT - is used for times. At midnight," at noon", Or contexts ex: " at work, "at school" especific places .

You may use both 'at' and 'in', depending on the context: 'She bought it at the chemist's'. 'I work in a chemist'. 'I'm in the drugstore' (emphasizing you are inside the store, not outside). Here's an example from Cambridge Dictionary: 'Jane was a pharmacist in a chemist's in a small town'.