If your kitchen is making miasma it means your food is rotting - make sure you have enough food stockpiles and barrels.
If it's a butcher's workship that is making miasma, it could also be hide (which can be made into leather at a tanners workshop) - a refuse stockpile will hold hide (and other by-products of butchering like skulls and bones) if you don't want to make it into leather.
A bit confused by your first question, but hopefully some of this will answer it:
* if you (d)esignate up stairs or up/down stairs, they will be dug out of the rock and will fill the tile they are in. If you want to build something else where they are, you need to use the designation 'remove upward ramp/stairs' first.
* if you (d)esignate down stairs, they will be dug out of the floor but will not obstruct regular buildings. They will only obstruct constructions that use the floor (like up/down stairs or down stairs).
* if you (b)uild some (C)onstructed up stairs or up/down stairs, they will be made out of the material you choose and will fill the tile they are in. If you want to build something else where they are, you need to use the designation 'remove construction' first.
* if you (b)uild some (C)onstructed down stairs, they will be built in place of the floor. If you want to build something else where they are, you need to use the designation 'remove construction' first.
The main things to remember are that if you have already mined out a tile, you can only dig a downward staircase there. If you want an upward or up/down staircase, you will need to build one.
Likewise, if you have dug an upward or up/down staircase, you cannot build anything in that tile unless you remove the 'upward' part. This will leave you with an empty floor or a downward staircase (which can be built over), depending on the type you started with.
If you have dug or built an upward staircase and want an up/down staircase instead, you need to remove the upward staircase (with the appropriate designation) AND channel out the floor (again, with the appropriate designation) before you can construct an up/down staircase in it's place. You can actually skip the remove step if you dug the staircase, since channelling will do that for you.