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I don't think so no. I like plants for a more natural tank, filtering water a bit and a steady diet of roughage for the fish so that all says it would help with stocking. However I fully stock anyway and with plants coming and going, I prefer to play it safe.
Yes and no. Yes you can if you have a forest growing in your tank, by that I mean heavily planted. You would be able to stock more fish because the plants will suck up the heavier bio load but that would not be the case if you just started the tank and planted it.
It will only have a little impact. Like others have said the plants will absorb the nitrates but even with a heavily planted tank, you may be able to add 1-2 more small fish. It will not make a drastic change. However real plant make any tank look way better than fake plants.