Anche se la giornata svanisce lentamente, il profumo del parco resta nella mente.

anche se is a conjunction meaning even if or although. It introduces a concessive clause and is normally followed by the indicative mood:

• anche se piove, usciremo lo stesso.
  By contrast:
• sebbene and nonostante (meaning although and despite) usually take the subjunctive or a noun phrase:
  
  • sebbene sia tardi, continuo a lavorare.
  • nonostante la pioggia, usciremo.

Yes, both mean “day,” but with a nuance:

• giorno refers to a 24-hour period or a calendar day.
• giornata focuses on the experience or course of daylight hours.
  Here, la giornata svanisce evokes the passing of the day’s light and atmosphere more poetically than il giorno svanisce would.

svanisce is the 3rd-person singular present of svanire (“to fade,” “to vanish”). svanire belongs to the -ire verbs that insert -isc- in all forms except noi and voi:

• io svanisco
• tu svanisci
• lui/lei svanisce
• noi svaniamo (no -isc-)
  The pronunciation is /svaˈniʃʃe/.

Most Italian adverbs ending in -mente come from the feminine singular of an adjective plus -mente:

• lento (m. adj.) → lenta (f. adj.) → lentamente (adv., “slowly”)
  You can do the same with, for example, veloce → velocemente (“quickly”).

In Italian, di + il contracts to del. This is called an articulated preposition. Other examples:

• a
  • la → alla
• in
  • lo → nello

Both restare and rimanere mean “to remain”/“to stay”, but:

• restare often feels more static or poetic (“to stay put”), and is common in literature.
• rimanere can imply continuation or what’s left (“what remains”).
  Here, il profumo del parco resta gives a slightly more elegant, lingering tone.

in + la → nella, so nella mente literally means “in the mind.”
You can also say avere qualcosa in mente (“to have something in mind”), dropping the article.

• resta nella mente (“it stays in the mind”)
• ho un’idea in mente (“I have an idea in mind”)

The present tense conveys a timeless, poetic truth: the fragrance always remains in the mind, even as the day fades.
You could use the past for a specific event:

• …la giornata svanì lentamente, il profumo del parco restò nella mente.
  But that shifts the statement to a once-off, historical narration.

Yes. You can place the main clause first:

• Il profumo del parco resta nella mente anche se la giornata svanisce lentamente.
  This just shifts the emphasis slightly onto the fragrance before introducing the fading day. Both orders are grammatically correct.