Before you bind the mesh to the armature (parent With Automatic Weights) you need to apply the Rotation of the mesh. Your character was modeled lying on the floor and then rotated by 90° to get it up. When you bind it to the armature it's reset to the original position on the floor. You need to apply the rotation to tell Blender that the standing pose is the new original position.
Select the mesh in Object mode, then use Ctrl+A > Apply All Transformations. This applies Scale, Rotation and Location.
To have everything nice and clean you can do this for the armature, too. Then parent them together.
If some loose parts of the mesh such as the eyes and teeth don't follow the armature then the automatic weights didn't work correctly. This happens often for loose parts. You need to assign them to the head bone manually as described in this question here: Tongue and teeth don't follow the movement when I change the pose, any suggestins on how to fix this?
Update after inspecting the blend file
There are several problems.
The rig (armature object) is not ideal. Your model is 90 meters tall but all bones are tiny and aligned vertically. I don't know how to fix this easily. The bones for the eye are overlapped by the upper and lower eyelids for example. It's not very comfortable to use and I am not sure if automatic weights work well with these tiny wrong aligned bones. Maybe consider using Rigify to create a new rig?
There are two Armature modifiers. Remove the second one to fix the teeth and eyelashes.
Furthermore, the mesh is not aligned properly. You need to move it a bit to the left (←).
For the left eye (on the right side) the vertex group acts weird. Even if you use the Remove From All Groups button there are some verts of the right sleeve moving along. I've deleted the group, created a new vertex group with the same name, and assigned all vertices of the eye to the group. Then it worked.
The mesh has a lot of loose parts. Automatic weights don't work very well with loose parts. If you parent the mesh to the armature (With Automatic Weights) you will notice that there are problems with the feet. And around the area of the belly/back, there is a hole when the character bends. This can be fixed with a remeshed duplicated dummy model that is used for the automatic weights. Then the weights are transferred back to the original model. This method is described here: heat weighting: failed to find solution for one or more bones or check this video.
If you're new to rigging then it might be easier to start with Rigify or a model that is already rigged like the Rain or Vincent. Vincent uses BlenRig. If you want to try BlenRig 6 beta for your character model you can get it here.
I've done the automatic weights and transferred them, and fixed the eyes. It works except for the legs. It's the bad rig with these tiny bones. It's the dots on the screenshot. It's not a custom shape but tiny sticks:

The mesh has not much geometry for the upper arms, and they bend not very well. The rig has no IK. That's another disadvantage when you want to animate the character.
Blend file: https://pasteall.org/blend/6d5634b7e22a4d93abf21dd2713039fb
Alt+A=unselect all),then select one vertice and pressLto select the eye. Keep the mouse over it, then Remove from All Groups, then Assign. Repeat the steps for the teeth – Blunder Oct 04 '21 at 00:39