There's an error in these lines:

    localRotation.y += Input.acceleration.x * speed;
    localRotation.x += Input.acceleration.y * speed;

localRotation is a quaternion, and their x,y,z,w components have nothing to do with the familiar angles we see in the Rotation field in the Inspector - this field is actually transform.eulerAngles. It's a very common mistake, which most people do at first (me included).
In order to keep your logic, you should declare localRotation as a Vector3, and store transform.eulerAngles in it at Start - like below:

using UnityEngine;
using System.Collections;
public class Playermovement : MonoBehaviour
{
    public float Speed = 16.0f; //left/right movement speed
    private Vector3 localRotation; // declare localRotation as a Vector3
    public float speed = 16.0f; // left/right movement speed
    public float rollSpeed = 5.0f;
    public float rollAmount = 10.0f;
	
    void Update() {
        rigidbody.velocity = new Vector3(0, rigidbody.velocity.y, 0);
 
        transform.Translate(Input.GetAxis("Horizontal") * Time.deltaTime * Speed, 0, 0);

        Vector3 dir = Vector3.zero;  //Android controls
        dir.x = -Input.acceleration.y;
        dir.z = Input.acceleration.x;
        if (dir.sqrMagnitude > 1)
            dir.Normalize();
        
        dir *= Time.deltaTime;
        transform.Translate(dir * Speed);

        // first update the current rotation angles with input from acceleration axis
        localRotation.y += Input.acceleration.x * speed;
        localRotation.x += Input.acceleration.y * speed;
        rollAmount *= 0.9f;

        if (Input.GetKey("left")) {
            rollAmount += rollSpeed ;
        } else if (Input.GetKey("right")) {
            rollAmount -= rollSpeed ;
        }

        transform.Rotate(0, rollAmount*Time.deltaTime, 0);
        rigidbody.freezeRotation = true;
    }  

    // Use this for initialization
    void Start (){
       // copy the rotation of the object itself into a buffer
       localRotation = transform.eulerAngles;
    }
}

But you're modifying the Z position in the "Android controls" section, as @You (not you!) said - maybe you (not @You - God, this is becoming very confusing!) should use dir.y instead of dir.z in that section.