You’re “tapping a permanent for mana” only if you’re activating a mana ability of that permanent that includes the {T} symbol in its cost. A mana ability produces mana as part of its effect.

If an ability triggers “whenever you tap” something for mana and produces mana, that triggered mana ability won’t be affected by Nyxbloom Ancient.

Nyxbloom Ancient doesn’t produce any mana itself. Rather, it causes permanents you tap for mana to produce more mana. If the mana ability of that permanent puts any restrictions or riders on the mana it produces, that will apply to all the mana it produces this way.

The effects of multiple Nyxbloom Ancients are cumulative. For example, if you have two Nyxbloom Ancients on the battlefield, you’ll get nine times the original amount and type of mana.