Game Movement Howto?

Hello,
I am after an time again in game dev and I have an question about the movement of entities.

The game is orthogonal and with tiles.

I have implemented A* which searches an valid path to an mouse click. And i am operationg on nodes=tiles. Where I don´t knew if it´s very intelligent. So I have an path object.
I don´t knew how to move then.
Normal keyboard input is working.
I am just multiplying the world.delta with the velocity and add it to position.

1.) Movement from tile to tile
lets say i am only going from Tile to Tile. How should I do this if I don´t want that an step happens after one tick?

2.) pixel movement
When in my Path are only “the tiles to go” stored, how should i move pixelish?
Sure, I knew the tileSize.
But I don´t knew how I should go automatically.

The only idea I had was:
get the Node
calc it´s pixel value
set the velocity to the max speed
and then go to the next position
check if outside the node
begin again util list empty
if node.pixel != current.pos -> go util click.pos

But I have the feeling that there is something missing and that it´s not an very nice way todo so.
And how sould I go if one tile is partly solid?
Let´s say my spirte is 32x64 pixels and my tiles 32x32.
A* calulated an valid path, but if I go there my sprite will collide with an object, because it´s above the valid Tile.

I hope it´s understandable what I wrote…