Seems you have two seperate problems…
In the first post the chain rubs on the front mech due to lack of tension.
In the second post, your chain is getting caught up when changing from middle to small ring, this is chainsuck.
Have you only been getting chainsuck since you put the new chain on? If yes, thats due to the drivetrain being too worn for a new chain, two solutions: Replace everything (£££) or put the old chain back on and ride it for another year or two until it gives up. Adjust the front mech down slightly to stop the chain rubbing.
If you had chainsuck before as well, then it could been the same problem as the chain rubbing on the mech – lack of tension. Lift the back wheel off the ground and spin it. If the pedals spin as well, the freehub is probably the cause. If not, its going to be either stiff jockey wheels or the main spring in the mech. Or it could even just be that the mech/spring are caked in mud and just needs a clean 🙂