I mulch my thyme in my own herb garden with straw, but haven't in a demonstration garden (but plan to this year)

As for how closely to the roses they should be planted, I'd recommend at or past the roses drip zone. I planted silvermound artemisia around my roses at this location and every spring thinned it out so it didn't invade the rose drip zone.

I mulch mine with wood chips, but I've seen big ones that have never been mulched. Mulched ones do seem to come back faster in the spring, though.

mulch thyme? Hmm... I don't, I live in a hot zone and no mulch... but then I water every 3 days... I should probably mulch.

Our summers are hot and dry and my soil is quickly draining...sandy/gravelly. Both creeping thymes and German Winter Thyme are mulched with landscape gravel only in full southern or western exposure and plants remain evergreen. They also do not demand a lot of watering. I would be afraid to have any organic material as mulch around them especially in late fall and spring during wet weather as thyme doesn't like too much moisture near the crown.
BTW organic mulches (stray, leaves, bark, etc.) in spring do not allow plants to 'come back quicker'...they soil is not allowed to warm as quickly. My Thyme, Lavender, Salvia to name a few are actively growing by March-April. The beds that do require mulching don't get their straw until June or whenever the soil warms enough and plants are leaping. This spring will be much earlier with the above normal current temps...it's been 90+ for the last 3 days! Arghh.

Vera

I've bought some too, and wondering if it's readily available at nurseries

Creeping thyme https://www.ehakimji.in/product/veechem-ashok-twak-sabut-50-gm-pack-of-1is a hardy herb primarily grown in the home garden for its decorative aspects. Its creeping habit may bring the question: Is creeping thyme invasive? While winter hardy, the short answer is no. It grows well in U.S. Department of Agriculture plant hardiness zones 4 through 8, though it may tolerate hot summers in zone 9 as well. Although you wouldn’t plant creeping thyme with mulch, you can use mulch in the wintertime to prevent frost or freeze damage.