I will break this into parts as one is very simple.

NOTE:  I have played a LOT of Crusader which has many types of equipment to obtain so there is a fair degree of ease it seems as you level to find things you can use. Psyker is a lot more restricted with equipment choice once you shape your abilities (though as a class it offers a lot more diversity in build choices). 

My Psyker is level 57 now and I have 2 crusaders (1 melee and one range) at 77 and 72 so I may have some bias in my observations about item drop rates for type of drop but believe my observations about drop rate are accurate.

1. In Tarot Missions which card increases chance to get single handed warp rods for psyker? Is it pistol? 

2. when Game was released relics were rare as we leveled but once we were over the item cap (I think it was 40 with player level cap 50?) relic drops were far more common - especially in the tarot missions. So...levelling was a challenge and I recall 30-40 being a tough grind due to needing upgraded gear but you had the great chance to improve your gear over the last 10 levels of levelling as you were over the item level cap and relic drops seemed more common. 

Player Level cap is now 80, and item level cap 70. 

I have to level 70 levels before I access that increased relic drop rate. I am finding I can do 6-8+ Tarot missions at +250 and get no relics when playing my level 55 psyker but on my Level 78 Crusader I get relics easily.

I do not know if this is by design but if my experience is the way the game is functioning it will make item diversity and relic opportunities progressively worse as you continue to release new content and increase the level cap. 

As an example - after 57 levels I have one relic warp rod and it was a level 18 given to me by a cabal member.....

I mention this NOT as criticism - but to highlight my experience so you can reconfirm that the relic drops are functioning as the Development team want. 

3. Seperate to this is the relic drop rate in Warzones. Not all player builds can do warzones though and you need to be quite high level to get those high rates of relic rewards. Great for my melee crusader :D but not my squishy psyker or assassin.