H4831, IMR4831, Re22 and IMR7828 have all worked well for me in 7mm rem mags with various bullets. I use Re22 the most now as it seems to work well with 140 -160 gr pills. Like said giving you a starting or good charge level is impossible as we don't know if it is safe or will work in your rifle, so get some references and work up slowly. Change only one component at a time and track the results till you find loads of interest. I use .5 gr changes once I find loads that show promise, till then I usually run 1 gr increminets when load developing.

As far as bullets the list is pretty long for your requirements. Honestly Nosler has always been my go to in terms of bullets, they have always worked well for me on antelope thru moose. However others such as hornady, barnes, etc have also worked just depends on what I am hunting, terrain, time to develop, etc. I'd suggest looking at Nosler Part 140-160gr, Nosler Accubond 160gr, Barnes TSX BT 140 or 150gr. Other such as 154gr InterlockSP or Interbond by Hornady, swift aframe, CT Failsafe, etc are also options for you.

Good Luck and enjoy.