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Is there a way to code a card (say, "Birds of Paradise") so that it can use its mana abilities on the turn it comes into play, but is otherwise affected by summoning sickness? Or is there a way to make a granted ability usable on the turn a creature comes into play without removing a creature's summoning sickness? If so, what might that code look like? Thanks for all your help.

There is a way, but that is not how magic rules work. The rules state that you can't use abilities with in them the turn a creature comes into play unless they have haste. This also applies to mana abilities, so as things are currently coded it is as close to the actual rules as we can get.

That being said, if you really want to code a activated ability (also applies to Manual Mana abilities) so that it can be used the turn a creature comes into play without haste (so that summoning sickness still applies) you can remove TapSelf from the cost and put EffectSource():Tap() into a PLAY_TIME_ACTION and add an AVAILABILITY block to the ability to prevent activation if it is already tapped. Though if the TapSelf is the only cost then you will have to replace it with another cost (a mana cost of usually works). The same would apply to granted abilities. This is just a method to do what was requested and is not compliant with MtG rules ( costs should be implemented as costs to be MtG rule compliant).

Thanks for the tip. I just tried the mana cost of zero, and it worked like a charm. Now I've got another question: is it possible to restrict the mana generated by an ability to only be used for certain purposes, as in Sliver Hive's second ability? More specifically, if I have a card with a mana ability, and I want the mana it produces to only be used to cast cards with the name of said card (or to pay the casting cost of said card), is that possible?

No, I tried to make a way to restrict mana, but I was fighting with just too many limitations and bugs in the engine. There is no way to restrict mana usage.