I think your confused with this method ;)
First, to activate the Mystery Gift function, go to the Jubilife TV Center..on the second floor, there is a man who accepts sayings from you. Tell him the following; "EVERYONE HAPPY" & "WI-FI CONNECTION" and he shall say he can pull some strings and get your Mystery Gift. Save your game and reset
On the main menu now will be the Mystery Gift feature. Mystery Gift has several options for you to utilise; "Get From A Friend", "Get Via Wireless" & "Get Via Nintendo Wi-Fi Connection"
Here Is for more Info
I love Weird things ¯(°_o)/¯
The Place Should Look Like This BTW
Then here is the Mystery Gift Option screen for Help ;)
To naswer your 2cd question simply NO that isn't true ;)