You can install your new fly reel so the reel handle is on the left side.
There are probably a few mods you must do inside the reel in order to do
this.

Yes, the fish is played from the reel. As you know now, a cheap reel can
cause problems.

You should have dacron backing on your reel, under the fly line.

A fly reel usually has no drag, your palm against the rim of the reel
applies the drag.