In 5E can you customize weapons or armor?
Yes, you can - but that doesn't make the weapon or armor anything other than what it is.
Take a Shortsword for instance, it's a 1d6 melee weapon with Finesse and Light attributes.
You can flavor customize it and say it's a really well crafted fileting knife if you want or a gladius.
The stats stay the same.
We keep finding gems everywhere.
Gems are common in D&D, but they also have a use in the core rules other than for selling - BG3 seems to have removed Components for spells all together, making gems just another source of gold instead of something you would hoard in case you need to cast a spell.
And can feats be earned any way other than leveling?
With the exception of the Human Variant Race, no. That said, after level 20 you can get something similar to feats depending on the mood of your DM.