And lernu says that the correct translation for giggle is "nervoza rido" [nervous laugh], "embarasa rido" [embarrassed laugh] or "koketa rido" [literally little chicken laugh, but I'm guessing it's more accurately translated as coquettish laugh]. None of these seems to quite capture the connotations of 'giggle' to me, but I'll definitely be using the last one!
I might have suggested "kaŝridi" but both J.C. Wells' & Peter Benson's dictionaries agree with "subridi". However, context does play an important role when looking for just the right word. Is the sound or the motive more telling for your particular use? The answer can definitely dictate which word you choose.
May I offer some slight corrections? It should be "nervoza rido" with a zed. Also, be careful of your grammatical endings; remember that the adjectival a-ending affects nouns, but the adverbial e-ending affects verbs. It's correct to say either "embarasa rido" [an embarrassed laugh] or "embarase ridi" [to laugh embarassedly] and either "koketa rido" [a little chicken laugh] or "kokete ridi" [to laugh like a little chicken].
I was amused by your translation of "koketa" as "little chicken." There is an Esperanto root "koket-" which is a cognate of the English word "coquette" and has nothing to do with chickens, little or otherwise. The phase "koketa rido" is a coquettish laugh, that is, a giggle. This sometimes happens in Esperanto, where a root may appear to have a different meaning if you interpret parts of the root as affixes. This can be used to good effect in jokes.
Well interestingly, the English word 'coquette' is derived from a word that originally meant chicken! The French word 'coq' (rooster, cock) became the diminutive 'coquet' (male flirt), which became 'coquette' for a female flirt. In other words, a coquette is a little chicken ;)
In essence, they both serve the same purpose as "please" but they are grammatically quite different.
"Bon-vol-u" is an imperative (command) verb which roughly equates to "Be so kind." Most often it takes a subordinate verb with the infinitive tense, as you saw above. A literal translation of it is: "Be so kind as to smile for me".
"Mi petas" is a complete sentence meaning "I request". You could expand this to be "Mi petas ke vi ridetu" (I ask that you smile), but it's often just tacked onto the end of a command, thus: "Ridetu, mi petas" (Smile, please). If you were ordering something you could be explicit and say "Mi volas mendi unu kafon" (I want to order one coffee) or you could just leave everything off but the subject, and say "Unu kafon, mi petas" (One coffee, please).
It's because bonvolu is already in the imperative, and you can't have two imperative verbs in a row. If it was just "Smile for me" (instead of "Please smile for me"), it would be "Ridetu por mi." Or you could use mi petas instead if you wanted the keep the please in there: "Ridetu por mi, mi petas."