I hope I posted this on the right forum selection.

Not sure if it's a bug... I gave her the stronger bow but she still uses her default bow. Do I need to give her better arrows than dwarven and steel arrows she's currently carrying for her to be able to equip stronger bow? Thanks.

Companions always equip the strongest gear they have, so there's something about her other bow that she likes better. Does she use the one you gave her at all and then switch or does she just immediately only use her default one?

Companions that use bow do tend to swap back to their default hidden bow, even when given a fully smithed, double enchanted daedric bow.. It's a bug. And theres no way to guarantee them using your given bow, but they will randomly equip it and swap back to default the next fight.

Lydia would never equip a stronger Soul Trap enchanted bow for me. If I gave her an unenchanted, smithed Legendary Imperial Bow, she would not equip it either. If I gave her a fire enchanted, smithed Dwarven Bow, she would use it; but not one enchanted with Soul Trap.

Had an issue with her and arrows too. Kept defaulting to Iron Arrows, even when there were more than 12 of a much better type. Sometimes it would take, and I'd see her carrying the Orc arrows, and sometimes it would be Iron, though she had plenty of other arrows on her.

She probably won't use Soul Trap because it tries to bind the soul to a gem owned by the caster, and the AI isn't set to do that. Try giving her some empty soul gems? I doubt that would work though.

Same problem here. Its not because of Soul Trap though, she just refuses to use anything but her default bow for me, which is a shame...never had this problem with my undead Bandit Chief thrall, he would swap all weapons (if he had enough and the right kind) multiple times in all fights, ie; using a shock axe against mages then his frost one for warriors, same with bows and staves, came in bloody handy in a tight spot, Lydia sadly is of little use because she always hangs back if possible and uses a crappy bow that does neglible damage. Just given her a few staves and hope she will start usig them instead of a bow.

I've had this issue with any warhammer. I gave her a Legendary Daedric Sword but if she has a basic Iron Warhammer she will switch to it (which means every time I'm looting areas and have her carry the heavy items). I've never had her use her original bow no matter what other bow I give her. Also with arrows, I never gave her any but she started using Ebony ones once I hit a certain level. I don't remember her doing it with my first playthrough though.

Companions are glitched, the Lydia bow glitch is KNOWN and CONFIRMED. It has nothing to do with her default bow being secretly awesome, it's terrible.

Followers, and enemy for that matter, will equip any arrows shot at them that get added to their inventory. Even when you get hit and after they fade some go to your inventory as well.

THIS ^
That's completely true. I've done that myself (minus double enchantment). And happens randomly. I even tried doing this with Aela the Huntress, since she is a marksman. She would equip the Daedric Bow in a fight, but 5 secs later, she would be back to her hunting bow that is NOT in her inventory, so you can't take that from her. It's just hidden, like her default armor. Arrows are equipped correctly though. Another [censored] game bug.