Every case is quite different…

First

Il faut accepter tout le monde peu importe (or “importent”, I have to check) sa race, son sexe, et sa sexualité.
Or maybe
Il faut accepter chacun, quels que soient …
I would use “N’importe qui” for “anyone can do the job”

Second

L’ours peut manger n’importe quel type de viande qu’on lui donne. (Not “que l’on”)
“Peu importe” would be used for “the bear can eat whatever hungry he is” “Peu importe qu’il ait faim”

Third

“n’importe laquelle” is correct.
“Peu importe ce dont tu parles, je ne le saurai pas”
Or “Quel que soit ce dont tu parles, je ne le saurai pas”

Forth

Before an adjective, I would use “aussi”:
“Aussi brillant que tu sois, tu ne deviendras jamais sympa”

I had the same thought as @Pierre-Louis Deschamps at reading your question, so I won’t repeat the same thing. Just to point this particular point :

Or is it that n’importe has to be followed by a word like “quel” or “laquelle” and peu importe can be followed just by a noun?

I’d tend to say yes, I can’t find any exemple of “n’importe” directly followed by a noun. But with “qu’importe” yes… (note the ‘qu’’) 😉

I guess this is because the verb “importer” (in its meaning “matter”, not “import”) is most of the time the verbe of the principal proposition, just here to introduce the subordinate…