If I put them outside in the ground now (NYC) will they grow now?

No. They will come up and bloom next spring.

Is the cold needed for all bulbs, but forcing (in the cold) only for those
you wish to grow out of season?

Yes.

What if it's still moderate temps for a few more weeks? Will the bulbs I put
outside grow now instead of in the spring?

No. They need several weeks of cold in order to bloom.

Would they grow now if I grew them in doors?

They need 14 to 20 weeks of cold preparation at 41 to 48 degrees, followed by
two to three weeks indoors to force them to flower. You can purchase
"pre-chilled" bulbs meant for forcing.
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