Normally we always need an article with adjective + determinate noun. With ordinals however, it is easier to leave the article out. I don't really know any exact rules for this, but it seems that it's more natural to leave the article out later in the sentence, but not at the start of a sentence. I can accept Hon är i den sjunde månaden and Det är den andra gången [and those answers are accepted there] although they are clearly better without the article, but if you skip the article in the above sentence, it sounds ungrammatical to me.
femte hjulet is more of a name-like concept, just like tredje mannen – I think those fall into the Vita huset = The White House category where we don't use the article for things that are names or have a status that is close to that.
For ordinals, only första and andra inflect – you can say den förste and den andre about masculine nouns (but you don't have to). The rest of these words only have one form each. So it's tredje, fjärde, femte, sjätte, sjunde, åttonde, nionde, tionde (3rd - 10th) and they all just have one form each.