Revised December 13, 2021 with new scores in the LRT.
Yes. And Yes.
The sandgrouse (genus: Pterocles) nests with pigeons in the large reptile tree (LRT, 1235 taxa then, 2018 taxa now), And those two do not nest with Tetrao tretix, the black grouse, nests between chickens and sparrows in the LRT. Pigeons as a clade nest as sisters to grouse as a clade.
Pterocles exustus (Temminck 1815) is the extant chestnut-bellied sandgrouse. This vegetarian prefers bushy, arid lands.
Timminck CJ 1825. Atlas des oiseaux d’Europe, pour servir de complément au Manuel d’ornithologie de M. Temminck. Belin, Paris 1826–42.