Il vento soffia dolcemente in giardino.

The pronunciation is roughly:
/il ˈvɛnto ˈsɔfːja doltʃeˈmente in dʒardˈdiːno/
– The double ff in soffia is held longer than a single f.
– soffia has a geminated f: /ˈsɔfːja/.
– giardino’s gli-like sound is /dʒ/.

dolcemente is an adverb meaning gently. In Italian many adverbs end in -mente, formed from the feminine adjective plus mente.
– Adjective: dolce (sweet, gentle)
– Adverb: dolce (feminine) + mente → dolcemente (gently)

Yes, you could say:
– Il vento dolcemente soffia in giardino. (more poetic)
– Il vento soffia in giardino dolcemente. (less common)
However, the most natural word order is verb + adverb: soffia dolcemente.

When talking about a general location without specifying “the garden” among others, Italians often omit the article:
– in giardino = in the garden (in general, as an activity)
Use nel giardino (in + il) when you want to emphasize a specific garden:
– Il gatto è nel giardino dietro casa. = The cat is in the garden behind the house.

Yes, soffiare is a standard ‑are verb. Present tense for soffiare:
io soffio
tu soffi
lui/lei soffia
noi soffiamo
voi soffiate
loro soffiano

Certainly. For example:
– Il vento soffia forte in giardino. (strongly)
– Il vento soffia piano in giardino. (quietly/slowly)
Notice that forte and piano already function as adverbs without the ‑mente ending.