I want to get the most out of the nettle and gotu kola herbs in tea form. If you put them in boiling water, don't you destroy most of the nutrients?

Yes, which is why I rarely use teas for medicinal purposes.  I generally drink my teas for flavor.  Although there are some exceptions.  Pau d' arco, marshmallow root, licorice root, slippery elm, etc.  can all be boiled without damaging their medicinal properties.  Some herbs,  such as marshmallow root and slippery elm, are often boiled to make them more mucilaginous for coating the membranes.

Would it be better to soak them in warm (but not hot) water for a longer period so that you can maximize all the nutrients?

If you want the most out of herbs I recommend taking them as powders or in capsules.  Even without the heat there will be some loss of nutrients by the addition of water and letting it sit for any length of time.  Even tinctures generally have a very short life for the same reason.  Water present in most tinctures, often made with an alcohol/water mix, promotes oxidative destruction of some of the active components in herbs.

Although, there are times where a tea may be preferred.  For example if making a tea to treat a throat infection so it can be sipped to keep the throat coated.

Is that also true for licorice root and slippery elm?

No, these are best physically boiled.  Boiling extracts the licorice root more efficiently when using cut licorice.  And boiling slippery elm helps to convert it in to a mucilage.