The OED entry for cobbler, which isn’t among the fully updated ones, says of the iced drink sense: ‘The origin of this appears to be lost; various conjectures are current, e.g. that it is short for cobbler’s punch, and that it ‘patches up’ the drinkers’. Dickens knew cobbler’s punch, which according to the OED was ‘a warm drink of beer or ale with the addition of spirit, sugar, and spice.’
It lists the ‘fruit dessert’ sense as ‘US Western’ but offers no opinions as to its derivation. The Wikipedia entry for Cobbler_(food) offers this:
Cobblers originated in the early British American colonies. English settlers were unable to make traditional suet puddings due to lack of suitable ingredients and cooking equipment, so instead covered a stewed filling with a layer of uncooked plain biscuits or dumplings, fitted together. When fully cooked, the surface has the appearance of a cobbled street. The name may also derive from the fact that the ingredients are “cobbled” together.
The OED doesn’t know the ‘reject cloth’ and ‘mummichog’ (thanks for the new word, I had to go and look that up) senses at all. But as two senses of the verb to cobble are ‘to mend clumsily’ or ‘to botch’, I suppose it’s possible that the textile sense arose because the cloth looked crudely mended, or just botched.
The OED gives one further sense: ’horse chestnut pierced and used for the game of conkers’. My guess is that this sense is related to cobble in the sense ‘pebble’, the more so as one of the quotations given for this sense says (from D H Lawrence) says ‘He pulled from his pocket a black old horse-chestnut hanging on a string. This old cobbler had ‘cobbled’—hit and smashed—seventeen other cobblers’.