I don't speak Swedish so I might be wrong, but I'm pretty sure this is correct.
I don't think sista veckan is a correct translation. The English word "last" can have two possible distinct meanings, depending on the context. It could mean either "final" or "previous." Swedish, however, uses two different words to refer to these two meanings. Sista means "final," while förra means "previous," so only the latter works in this case.
Yes. There are some adjectives that work this way, they sort of make the noun definite by themselves so that the front article is not required as it would normally be. In some cases, the noun will be indefinite, in some it will be definite. So we say nästa vecka but förra veckan. In some cases, either works. For instance we can actually say either på tredje våning or på tredje våningen 'on the third floor'. So the underlying reason is that the adjective already shows that the noun is definite, but how it works exactly varies from adjective to adjective. It's probably best to try to learn those on a case by case basis.