What are the FAI rules for circumnavigating the world?
"mins" linked to a set of rules at https://www.fai.org/sites/default/files/documents/sc_section_2_2013.pdf , I will try to interpret these in light of your question. I do not take any position on whether those are the official rules and I do not gaurantee my interpretation is correct. I have not read the entire document, just the parts that seemed relavent after searching.

There does not seem to be an official record for youngest flight around thw world. There are records for the fastest round the world (eastbound and westbound) and a circumnavigator diploma (not a record) for flying around the world.

Neither of them seem to have a restriction on the length of invidiual flight legs. Surprisingly other than the polar circumnavigations there doesn't seem to be any requirement to cross the equator.

The author of the route you posted gives a distance of 21738 NM which translates to 40258 km . The speed around the world (eastbound and westbound) records have a minimum distance of 36,770 km, there does not seem to be a maximum number of control points for this record. 

The requirements for the circumnavigator diploma are a bit different, there are a maximum of 20 control points which may reduce the measured distance, but the distance requirement is much lower for the circumnavigator diploma.

So this route seems to pass the critera for both the fastest round the world record and the circumnavigator deploma.