The optimal ‘loadout’ for Lydia

She won't use Elven stuff because it's weaker than her heavy type Steel armor, I guess because she's skilled in Heavy Armor and not in Light Armor. I've given her Dwarven stuff and she equipped it on the spot myself.

As for why she did equip the elven helm, well she doesn't have a helm at all in her default gear so any helm is better than what she has no matter the stats.

That is a bug associated with followers that already have a default bow in their inventory (this is usually the Hunting Bow).

... followers who start with any kind of bow in their inventory are bugged so that placing a different bow in their inventory will have no effect. It may look as if they equipped the new bow, but the follower will switch back to their default bow as soon as combat starts, even if that bow was removed from their inventory.

^{Source: http://uesp.net/wiki/Skyrim:Followers#Follower_Equipment}

I've noticed that at least several followers seem to prefer to use their own personal (non-inventory) hunting bow over at least some more powerful bows you might give them to use. So far the only way I've been able to get a follower to consistently use the bow I provided to them is to console remove the hunting bow SEVERAL TIMES until they stop equipping it. It took about 5 removed before it worked with Aela, using the command (targetted on her) removeitem 10e2dd 1.
...
For console-users that don't have the luxury of the in-game console, giving your follower a Forsworn Bow seems to be something of a workaround. It appears to be the only type of bow that totally overrides the default hunting bow in a companions choice of weaponry. It's certainly not the best bow in the game, but it's still a damn sight better, especially since it can be upgraded and enchanted.

^{Source: https://en.uesp.net/wiki/Skyrim_talk:Followers/Archive_3}

The Unofficial Skyrim Patch (original / Legendary Edition - PC) / (Special Edition - PC) / (PS4) / (Xbox One) should fix that bug. Related patch note:

USKP v1.0 - (2012-04-07)

All followers will no longer have invisible Hunting Bows that can't be removed from them that they would use even in preference to much better bows they were given; they will now be visible and can be removed from the follower normally.

Other mods (for the PC) that can also fix the bug are: Follower Hunting Bow Fix, DFB - Disable-Show Follower Bow And Arrows, or Followers Hunting Bow Replacement. The followers overhaul mod, Better Followers also allows you to remove the Hunting Bow from your follower, forcing your follower to use the bow you gave them.