Thank you for your note and your question.
Yes.  We offer a resizing service.  We can resize to 2 sizes smaller, or up to 2 sizes larger on most rings.  Up to 3 sizes larger on some rings.  This service usually takes 2 - 3 business days depending on workload.  Our current charges are:
Yellow & White Gold
-Up to 2 sizes smaller:  $22.00
-Larger 1 size:  $34.00
-Larger 2 sizes:  $39.00
-Larger 3 sizes:  $47.00

Other services:
-White Gold Items (ring, simple bracelet) Clean & Polish:  $24.00
-Yellow Gold Items (ring, simple bracelet) Clean & Polish:  $18.00
-White & Yellow Gold Add Sizing Beads:  $40.00

Note:  Let us know using our contact form if you require any of these services.  We will provide instructions for sending us your item.  For purchased items, resizing or polishing jewelry voids/negates our 14-day return option and the item is non-returnable.

There are over 200 countries around the world and each have different shipping and postal regulations.  For Canada and the UK, for example, we can usually ship for free as well.  For other countries different fees are required by the destination countries and we are required to charge those fees. 
Shortest answer, No, international shipping overall is NOT expensive and in many cases is free as well.  Please use the Form on our Contact Page to inquire and we'll quote the lowest price possible.

Lucky you to be born in April.  The birthstone for that month is Diamond.

Below is a quick reference to birthstones from GIA.
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We're also pleased to provide this clickable link to the Gemological Institute of America (GIA) for their GIA BIRTHSTONE CHART and an explanation of the various stones.

We do, but as you're doubtlessly aware, Alexandrite is very expensive.  Even in its synthetic form.
But if you would like just use the Form on our Contact Page to let us know that you're looking for this item and we'll add it to our buying profile.  Should we come across an Alexandrite ring we will notify you, provide you photographs, and quote you a price.  All for no charge whatsoever.  If you like the ring, you simply conclude the purchase.  If you don't like the ring, we'll just add to our inventory.  Again, no charge to you whatsoever.  And...should you purchase the ring but upon receiving it find it not to your taste, just return within two weeks for a full refund.  
​Just let us know if we can be of service.  Aloha.