The fireplace that the letters came through in The Philosopher's Stone was in the kitchen:
Something came whizzing down the kitchen chimney as he spoke and caught him sharply on the back of the head. Next moment, thirty or forty letters came pelting out of the fireplace like bullets.
Harry Potter and the Philosopher's Stone - p.35 - Bloomsbury - Chapter 3, The Letters from No One
This fireplace was clearly not boarded up.
However, the Weasleys managed to turn up behind the fireplace in the living room:
From the other side of the living-room door came the sounds of the three Dursleys scrambling, panic-stricken, across the room.
Loud bangings and scrapings were coming from behind the Dursleys' boarded-up fireplace, which had a fake coal fire plugged in front of it.
Harry Potter and the Goblet of Fire - p.42 - Bloomsbury - Chapter 4, Back to The Burrow
This is a totally different fireplace to the one in the kitchen, which remains unobstructed into the days of The Order of the Phoenix
WHOOSH. With a clattering, a whirring of wings and a soft fall of dust, a fourth owl came shooting out of the kitchen fireplace.
Harry Potter and the Order of the Phoenix - p.36 - Bloomsbury - Chapter 2, A Peck of Owls
How did the letters get to Number 4 anyway? I mean, the first letter probably was transported by the normal post, but as there is no post on Sundays, this can't have been the case later.
(Strictly speaking it's one question at a time on this stack, but as it's a one-word (or one-name) answer ...)
'I'm - er - not supposed ter do magic, strictly speakin'. I was allowed ter do a bit ter follow yeh an' get yeh letters to yeh an' stuff - one o' the reasons I was so keen ter take on the job -'
Harry Potter and the Philosopher's Stone - p.48 - Bloomsbury - Chapter 4, The Keeper of the Keys