1. Double fusing isn't permitted in most countries, but not an issue for you anyway.
2. He's referring to X and Y rated noise suppression capacitors. The X capacitor is there in to reduce differential mode noise emissions across the P/N. It is placed across the input, hence it's X (normally X2) safety rating. The diagram is a little unusual, it is normal to have (in sequence from mains input) Fuse, X cap, Common mode choke, Bridge rectifier. It is possible that C1 is intended to be the X cap (non polarised film), I can't say without component values. L1 and BJ1 would normally be combined as a common mode choke too (to reduce common noise emmisions).
Assuming that you have a mains ground connection, The Y rated safety caps would be between P and mains ground and N and mains ground. It is normal to put a further Y cap between primary and secondary windings of the transformer to shunt switching noise. X caps are self healing (normally polypropylene film) capacitors which are safe for use directly across the mains without risk of dead shorts or fire. Y caps (normally Ceramic) have a breakdown test voltage so high (several kV) that they count as reinforced insulation, as failure would constitute an electrocution risk.
One other thing I noticed, on the primary side, several points go to a downward triangle symbol. Note that this is does not ground or chassis, it should be considered as a circuit common connection and not connected to either.
Edit: It's also normal to have a VDR on the input (again, after the fuse) to clamp damaging mains transients.