Used after a noun to suggest an example or a tentative choice, implying there are other possibilities. Often softens a suggestion.
How about some tea, for instance?
Would you like something to drink, like tea or coffee?
... or ...
Used in the pattern ...なり...なり to list options, often implying that either choice is fine or that the listener can decide.
Please contact me by phone or email.
Follows the dictionary form of a verb. Indicates that the second action happens immediately after the first.
As soon as I got home, it started raining.
while still; with the previous state continuing
Follows the past tense of a verb. Describes an action done while the result of the previous action remains unchanged.
He sat down and wouldn't move, staying just as he was.
Please contact me by phone or email, whichever is easier.
The moment he got home, he collapsed.
The shoes were left right where he'd taken them off at the entrance.
なり in the 〈物〉なり sense politely suggests one example among options, while とか is a casual, everyday way of giving an example.
なり (dictionary form + なり) means an action happens the instant another finishes, while 次第 (verb stem + 次第) is the more formal, written equivalent.
なり in this sense (past verb + なり) describes an action left in its resulting state, similar to まま but rarer and more literary.
Derived from the classical Japanese copula なり, which itself developed from にあり. The particle uses evolved from its function as a copula.