There have been occasional incidents involving yachts on the Rio Dulce - and certainly many issues within Guatemala - that have produced bad press for some years now...but they typically either occur in remote areas of the lake (up above Fronteras) and/or involve unoccupied boats anchored out on the Rio. I would classify the area of the Rio adjacent to Fronteras, where the vast majority of the yachts congregate, as far safer than VZ but a bit riskier than the DR, and not a place to avoid due to its threats. It''s the only option during hurricane season, which means you''ll see lots of other boats.
You of course can go where you wish at any time of the year, but being in the W Caribbean during hurricane season is like rolling loaded dice. Other than up the Rio, you ultimately can be at the mercy of any major storm that rolls thru there, and I''d encourage you to avoid it. Belize, Mexico and Honduras have all been severely vacuumed up by hurricanes in recent years, and you only have to look at the denuded hills in the Bay Islands to realize how frightening it must have been there when one hit them 3 years ago.
Paperwork isn''t a problem. The ship''s document, passports, a good look at the State Dept''s website on innoculations recommended (especially relevant if/when you travel inland in Central America, a wonderful experience) and a decent budget will open all doors. Rauscher''s guide is a ''must'' and we found Calder''s guide, while dated now, to offer an excellent discussion about wx, routing choices and the other detail you are asking about re: how to get there. The Tortugas are a favorite stop along the way, but you may want a direct run with a good stable f''cast if leaving from further up FL''s west coast. Be sure to download as current a pic of the Gulf Stream as you can before departing; while it changes hourly, having a decent idea of where the main Gulf feeder stream is (the one heading N into the Gulf from the W end of Cuba) and knowing the location of its cold & warm eddies can make a big difference in how fast you run over the ground, and to some extent on how comfy it is, too.