Fresh Thyme is a hardy, woodsy herb that releases its flavor best when boiled/cooked. Same advice for Rosemary and Sage. If you're going to use it fresh, add it to the boil. You could also add a secondary addition. HOWEVER, for a secondary addition, I would not add the actual herb, but rather a dash of concentrated Thyme extract. You can purchase it or make some at home.
If you were going to use the more delicate fresh herbs, such as Parsley, Cilantro, Basil, or Chervil, then the recommendation would be to avoid heat and solely use the fresh herb for best results.
For a complementary hop:
Pacific Jade (NZ)
This variety is the most recent release out of the New Zealand Hop Research Program selected on its average alpha of 12-14% with Cohumulone in the region of 24% and an excellent oils profile. The aroma of this hop is described as “bold” as it delivers a herbal infusion of fresh citrus and crushed black pepper. Brewing trials have illustrated Pacific Jade as an excellent hop that delivers a pleasing soft bitterness matched to desirable aroma characteristics.
Suited for use as a bittering hop with some excellent results also being seen in dual purpose applications, with a soft bitterness attributable to the low cohumulone. The citrus aroma and flavour notes work well to temper malt sweetness in “fullish” Ales especially when used moderately as a finishing hop. Pacific Jade is also well suited to balance dryer Lager styles when employed as an “up-front” kettle addition to showcase its bittering qualities.
Using Pacific Jade in conjunction with a mild yet aromatic European hop, like Saaz or Hallertau Mittlefruh may also provide some additional complexity.