If your eye get poked with a finger, there might be no problem. However, you should look in the mirror to have your eyes examed by yourself. If you get red lines on the white of your eye, or your poked eye feels painful when you move it around, your eyes should be examed by a doctor as soon as possible. If you find you get blurry vision in the injured eye, you'd better go to see a doctor immediately.
It depends on how hard the hit was. Usually, It will be fine if it not serve. Getting poked in the eye can very painful because our face have many nerve endings and the eyes are very sensitive. It may take a bit of time. It will be fine. If you want to make sure, you'd better let your eyes checked by eye doctor to see if you really hurt your eyes. Here online, there is nobody know can give your accurate answer since we don't know the condition of your eyes.
If your eyes just weep, they are nothing serious. This also is a kind of self protection. Tears can kill some bacterium. If there is nothing fall into the eyes, You can't rub your eyes and close them for a while. Many tears flow so that the dirty things inside can be rushed out. Or you can wash your eyes 2 or 3 times with 0.9% of physiological saline if having conditions. If there are some bloodshot on your eyes, you have to avoid using wrong eye drops, or it will be serious. But if your do not feel better after several hours, you'd better see a doctor.
That's doesn't matter. If you feel pain, you can close your eyes for a while. It will feel better. And if it is hurts, maybe you should see a doctor, he can give you some medicine, but after all, it is not a big deal, maybe you need some anti-inflammatory. You'd make your eyes have some rest.