Checking your pet is done in four ways. 1&2) : (shift colon=look) and/or #chat will tell you monster, tame/friendly or monster #chat will give you a hunger approx. if a pet that can communicate--see wiki). 3 a stethoscope will give some idea of things like invisible, confused, blind, etc 4 A wand of probing will tell you much more like equipment and carried items, Cancelled, etc. -- see wiki). I also am playing a Valk, and have a pet archon, I tried giving him a +2pair of gauntlets of power and +3Mjolnir, but for now he prefers a +4 leather glove and +7 RPLS. I will update if he changes preferences based on enchantment. (I really would like to see him throw that thing.)
Archons are naturally magic resistant and just get stronger in that respect. They can equip (as can many other monsters -- see wiki), you just need to know what they can use (ex; archons can armor up except for body armor, but they won't exchange unless there's a definite gain).Weapons are similar but there are quirks ( see my question about Djinnis and Demonbane).
Mainly, with an archon, ki-rin, or any other inediate ( they don't eat) monster, NEVER LEAVE THEM ON ANOTHER LEVEL, they will forget you fast.Archons are awesome until they turn on you. So, it's best to have a magic whistle, and use it often with these monsters ( I had a V Lord that I gave all my extras to, and fell down a trap door,. It took me 8 ticks on the time counter to get back up, and it went peaceful. Good thing I was chaotic... I never retamed it, but I wouldn't have wanted it to turn feral. Scrolls/spells of taming combined with some way of draining level (multiple times, re magic resistant) are needed for retaming.
If/when you get to Ludios, the castle, or find a barracks/throneroom, let it wander a while,. It will find the good stuff even if you don't. Be patient when giving a +2 to replace +1, they don't always like change.
As far as wishes go, consider yourself luccccccky, and do what Kevin says ( see the wiki for specifics) but don't waste another wish just to super it up(it's probably stronger than you as it is). Let it fight/win and level up, then maybe when it becomes resistant, get a blessed ring of conflict and use that and the whistle to get you out of jams like an arch-lich summoning nasties...
After that, an enchanted shield of reflection (if it doesn't already have), an amulet of lifesaving is a must ( they don't leave corpses to reanimate), and heavily enchanted armor (fixed enchanted non-metal --boots--elven if you have the scrolls to max with-- gloves dexterity maybe, but not really a boon, and helms HoBr if poss). Try to use a whip to steal its weapon, identify and enchant to return, or give it your 2nd best artifact/highly enchanted fixed Longsword). may Tyr be with you........