Why do we quote options in delta bid-ask & volatility bid-ask & why not it is quoted in terms of option premium?
Both the delta and volatility are a one-to-one mapping from the strike and premium respectively, so it does not harm to quote the latter in terms of the former. Now, does it any good? As options are often delta-hedged immediately, quotation in terms of delta allows to do so easily, cf. Wystup (link):
This allows market participants to ask various partners for quotes on a 25-Delta call, which is spot independent. The actual strike will be set depending on the spot if the trade is close to being finalized.
Quotation in terms of volatility allows to compare how over- or underpriced options on different strikes are, as it makes the in-the-money part of the premium be accounted for.
Fitting option pricing models to the volatilities makes more sense as well, as several stylized facts (e.g. smile, smirk etc) are only pronounced in this representation.