Relisting for me is great. I have multiples of most items in the shop, so every time I sell one, I will relist it to get it back into my shop.
As for renewing (spending an additional $.20 to extend the life of a listing by another 4 months) if I've got something that's about to expire then yes, I will renew it. However, I don't spend money on renewing items that are not about to expire. I would much rather add new items to the shop.
With the recent change to the search to relevancy rather than most recent, its more to your advantage to have good titles and tags than to throw money at renewing.
When I was married all my flowers were silk so if I'm looking for 'silk boutonnieres' I'll just put that in the search and see what comes up.
http://www.etsy.com/search/handmade?search_submit=&q=silk+boutonniere&order=most_relevant&ship_to=ZZ...If you notice, some of the items that show up on the first page of that search have listing dates from back in July sometime but because they are what I was looking for, they show up at the top of the search. I can now narrow down my search by flower type if I want to.
Yes, part of the search algorithm uses listing date, but if your title doesn't match what is being searched for, it doesn't matter what the listing date is.