Yes it is possible to mix Anthias species, I have groups of Lyretail, Ignitus, Resplendent, Sunset, Evensii and Bimaculatus living together with minimum aggression issues. However, the two species you have mentioned are arguably the more aggressive members of the anthias group, but the size of the tank and numbers of each are important considerations.
Depends on the species and the individuals involved. Bartlett and lyretails are more robust and can handle each other better, in terms of their assertive nature, however bartletts and have a reputation of all turning male and killing each other. I have 3 lyretails, 2 red saddle and 2 randalls anthias and they all get along well, and the lyretails were the first added and never posed any issues interms of killing the more timid species i have by being too assertive but it does depend aswell to an extent than the individuals involved. Typically mixing anthias with the same subgenus works, like the franzia genus containing lyretails and huchtii being the most agressive members of the anthids and the mibraulichthys being more peaceful and easier to mix as a general rule, so species like dispar and randalls