Yet another 2D pixel perfect movement question

I want to make a precise 2D platormer, something like Thomas Was Alone. The difficulty comes when trying to make a precise pixel perfect collision. I tried several different approaches, but none of them worked out the way I wanted.

These were my attempts:

1. CharacterController: I just couldn't adapt it to work in this 2D platformer. Its 3D's properties just caused me problems.

2. Moving the transform: some conflicts with the rest of Unity's physics because I'm moving the transform directly and not the rigidbody.

3. Changing the rigidbody's velocity: this was what came the closer to what I want, but it seems too unstable. The player keeps shaking and sometimes it bounces when falling.

In all these three options, I tried a lot of combinations with Unity's physics (like gravity, physics materials, rigidbody options).

The following code is my latest attempt (in this case, the game is a runner, so I need to move him to right every frame):

 using UnityEngine;
 using System.Collections;
 
 public class Character : MonoBehaviour
 {
     private float speed = 5.0f;
     private float jumpSpeed = 8.0f;
     private bool isGrounded = false;
     private bool isBlocked = false;
     
     private Transform myTransform;
     private Rigidbody myRigidbody;
     private Vector3 mySize;
     
     protected void Start()
     {
         myTransform = transform;
         myRigidbody = rigidbody;
         mySize = myTransform.localScale;
     }
     
     protected void FixedUpdate()
     {
         CheckCollisions();
         
         if (!isBlocked) {
             if (myRigidbody.velocity.x < speed) {
                 myRigidbody.velocity = new Vector3(speed, myRigidbody.velocity.y, 0);
             }
         }
         else if (myRigidbody.velocity.x != 0) {
             myRigidbody.velocity = new Vector3(0, myRigidbody.velocity.y, 0);
         }
         
         if (isGrounded && Input.GetButtonDown("Jump")) {
             myRigidbody.AddForce(Vector3.up * jumpSpeed, ForceMode.VelocityChange);
             isGrounded = false;
         }
     }
     
     private void CheckCollisions()
     {
         // Correction factor.
         float margin = 0.01f;
         
         Vector3 myPos = myTransform.position;
         
         isGrounded = Physics.Raycast(myPos, -Vector3.up, mySize.y / 2 + margin);
         isGrounded |= Physics.Raycast(myPos - Vector3.right * (mySize.x / 2 - margin), -Vector3.up, mySize.y / 2 + margin);
         isGrounded |= Physics.Raycast(myPos + Vector3.right * (mySize.x / 2 - margin), -Vector3.up, mySize.y / 2 + margin);
         
         isBlocked = Physics.Raycast(myPos, Vector3.right, mySize.x / 2 + margin);
         isBlocked |= Physics.Raycast(myPos - Vector3.up * (mySize.y / 2 - margin), Vector3.right, mySize.x / 2 + margin);
         isBlocked |= Physics.Raycast(myPos + Vector3.up * (mySize.y / 2 - margin), Vector3.right, mySize.x / 2 + margin);
     }
 }

Do you have any suggestions? I'll take anything!