Does sending 1 ant to a level 40 predator nest help increase your stamina?

Yes and no:-

Sending it to the nest won't increase your stamina but if you're at or close to your stamina cap, the 37 stamina will be taken from you when you start or join a rally and refunded to you when the rally is defeated.

If the rally time is set to 1 hour, you would be able to recharge stamina over the course of that hour.

For example, if you had 120 stamina and sent a 1hr to a nest your stamina would drop to 83. Over the course of that hour, your would recharge one stamina ever 6 minutes (or quicker, depending on research and talents). By the end of that hour, you would have 95 stamina and when the rally is defeated, the original 37 stamina is refunded. You now have 135 stamina, which is more than the normal maximum

NO, your stamina drops proportionately to the size nest you attack, aa said above. If someone joins the rally would be defeated and one ant would need to be healed. Your stamina refunded and you would gain any you generated. If Noone joins, it is disbanded and there is no attack. Mostly this is done in advance of an event that requires stamina. You can also repeat it hourly and even have two extra rallys going if you cap it again.