"I have not cooked since last week."
Translation:Yo no he cocinado desde la semana pasada.
If you say "this is the last week of the month", your suggestion would be all right, however, the expression "since last ..." is "desde la/el últim(o/a) semana/mes/vez ..."
you may also use "since the previous week", that would be "desde la semana anterior/previa)
Hard to know without seeing your whole sentence, but "desde hace" means "since ... ago". E.g. "desde hace una semana" = "since a week ago".
If you put that it means almost the same thing, but not quite. If, on a Monday I say "I haven't cooked since last week", then I might have cooked 3 days before, on the Friday. Whereas if, on a Monday, I say "I haven't cooked since a week ago" then I have not cooked since the previous Monday.
"desede hace" refers to a piriod of time. usually between a specified time and now, or with a specified length until now. and it does not imply nither that this period is ending nor that it does not ends.
Both mean "since", but "que" is needed when introducing a new clause (i.e. one that has its own verb).
"Yo no he cocinado desde la semana pasada" = "I haven't cooked since last week".
... as opposed to ...
"Yo no he cocinado desde que naciste" = "I haven't cooked since you were born".
Basically two slightly different structures...
Something has(n't) been happening (since) desde a point in time.
Something has(n't) been happening (since) desde que something else happened.