Translation:I think that dog is sick.
Quick usage note:
Don't make the mistake of saying びょう気です just because you have a かぜ (cold).
びょう気 is a rather strong word, and when I used it with a native speaker, she thought I was confessing to her that I was terminally ill. Needless to say, it was fairly awkward after that, and I only later learned why.
だ is the plain form of です. You need it to terminate the sentence in the 'that' clause. その犬は病気だ is a less polite form of その犬は病気です, and in colloquial speech you might hear people say そう犬は病気 and completely drop it. However, here it's a grammatical necessity; you must complete the sentence before saying と思います. その犬は病気と思います is grammatically wrong, even colloquially.
I'm a bit confused about だ. , in the notes for this lesson it says "The short past form is made by taking the て-form of the verb and replacing with た or だ."
So, is this だ a simple past? "I think the dog was sick" or it does not apply to です? I mean 泳ぐ (swim) short past form would be 泳いだ (swam), while だ still means "is".
I think the lesson notes are a bit confusing. Verbs have a て-form and a た-form (the latter of which is used for past tense). The conjugation rules for the two are identical. Thus, for -ぐ or a -ぬ verbs, て and た become で and だ respectively.
The "is" copula ( だ / です etc.) is unrelated. The past tense forms of the copula are だった and でした.