How to Ship Banknotes?

What is the best way to ship the notes so that they reach the destination with minimal or no degradation from the UNC state? (Without using any special envelopes or cardboard inside which will hike up the weight)

You need to use a larger rectangular square of cardboard than the currency sleeve.  The stronger (& thinner) the better.  I cut them to be just slightly smaller than my envelope (so the entire insert/note/insert doesn't move much).  You don't need a special envelope provided you use 2 cardboard inserts.  Auction houses use large cardboard envelopes + a cardboard insert.  (They're usually FedEx & expensive). 

For 2 cardboard inserts postage is slightly more but worth it.  It's slightly more expensive than regular postage for just the currency sleeve.  (I know this b/c I have a swap partner who uses almost nothing for our current polymer notes so I've used a currency sleeve plus a very light piece of cardboard stock & postage was regular postage- nothing extra).  I do not recommend this at all (especially for older/more valuable paper you wish to keep UNC).  If you wish to post VF notes (or have discussed it with your swap partner) than I suppose you should get away with it for circulated notes.

I use post-its & tape the non-sticky side to the cardboard insert & use the sticky side on the currency sleeve- to keep the currency sleeve (Note) in the middle of the cardboard.  I also put the currency sleeve (note) inside a plastic envelope (or wrap it with a small piece of saran wrap) in order to waterproof the note. I also use 1 extra (thinnest) cardboard stock insert just so nobody can feel that there's a currency sleeve inside my envelopes.  So the currency sleeve is sandwiched between 2 cardboard inserts (one sturdy, the other more for concealment).  

Hope that helps!

But will it clear the customs? Also, do the notes show when they scan or X-ray the package?

The more “commercial” a shipment looks like the more it will be scrutinized, at least that is my experience through the years. Printed address labels, padded envelopes scream “commercial”. I normally use ordinary covers (sized DIN C5) with stiffeners inside that does not make the shipment too thick, which is also a giveaway to customs. @Serial_Number_8 gives a good instruction on how to do the packaging.

Always add the text: “Do not bend!” and “Ne pas plier!” (French is the official postal language) to the front of the shipment. In case the shipment is bent, the recipient should, without opening the shipment, return the shipment right away by writing: “Reception refused” and “Réception refusée” to the shipment and hand it in at the local post office. In my country the postal service gives a compensation for a bent letter with this text applied to the front of the cover.