Streamlining both production and all the extra parts of running a business (customer service, shipping, product development, marketing, resourcing materials, etc.) is something we work on constantly - the more streamlined everything can be, the more time I have to do the parts of the business I really love :)
Etsy itself has really made some significant improvements in the last couple of years that help with this, and taking advantage of everything we can that Etsy offers (what works for us, at least) has been beneficial.
Saved Snippets, for instance, makes routine convos go a lot faster; this leaves me time to answer in detail convos that require complicated answers.
Etsy Shipping Labels saves us a huge (huge!) amount of time - I almost can't believe that when we started, we were shipping labels with stamps and handwriting the addresses! We got a Dymo, raised shipping $1 to cover First Class package rate (no one seemed to mind since shipping now includes tracking), and now shipping takes probably 1/4 the time it used to, or less - plus, since it's tracked, there are many fewer lost packages and inquiries about arrival times.
Promoted Listings takes care of all the marketing our shop can handle at the moment.
Being able to create custom orders right from convos saves us a ton of time setting up individual custom orders, and keeps the shop clear of clutter. Similarly, we don't have to make nearly as many custom orders as we did when we started because of variations with pricing included (we use variations in almost every listing, very efficient).
Expedited shipping options in the cart keep us from having to invoice separately for Priority or Express (huge timesaver); though it was better when customers could see these options, and the combined shipping costs, in the item listing.
Etsy doesn't take photos for you and I am a generally crap photographer, so I finally saved up and invested in product photography (just a few items at a time; half the shop is still my photography, but I'm getting there). It's a relief to create a new listing and know I don't have to spend 2 hours photographing and editing and rephotographing and cursing at the camera and etc. I actually list a lot more new items this way because I don't have to worry about taking the time to photograph.
Still, I don't have as much time for product development (which I *love*) and we're constantly on the verge of falling behind - one moderate illness and we're kaput. We've hired some part time help and that is good, but after Christmas I think we'll have to rethink how to organize the business to make things run more smoothly.