Translation:That park is pretty and quiet.
The problem with that, while understandable in itself, is that we have no context for any of the example sentences here. So it only comes off as inconsistency when in one case "the" makes your answer wrong, telling you it should be "that", while in another case you don't even have "that" as an option.
When you're linking adjectives with "and" the particle you use is で (for use after な adjectives) or て (for use after い adjectives). For example, if you want to say "She is quiet and strong" you would say 彼女おはしずかでつよいです because しずか is a な adjective. If you wanted to say "She is cute and strong" you would say 彼女おはかわいくてつよいです because かわいい is an い adjective. You drop the final い and addくて
It's not a particle! Please see my comment below. The particle で is used to convey an "instrument" used to perform an action はしで 食べます- I eat WITH chopsticks or a location 図書館で 勉強します- I study AT the library. The で used after na adjs which Tara mentions is NOT the particle で. It is the te form of です!!!
Yes, it is the te form of desu, and it acts like the conjunction 'and' and joins sentences together. To is used as 'and' with nouns - you would never use 'to' to join sentences. When you change the end of a verb so that it means 'and' and can link sentences together you will either use the te form of the verb or Base 2 of the verb eg. tachi (from tachimasu), shi from shimasu, asobi from asobimasu - Base 2 is basically the masu form of any verb minus the masu. Some examples of both gakkou ni itTE nihongo wo benkyou shimashita - I went to school AND studied Japanese. Kooen de asoBI arukimashita - I played AND walked at the park. In both examples I've capitalised the ending of the first verb to show what has been done so that it can act as a conjunction and join sentences. Hope I've explained this well.
Do you mean the order that you should translate? Of course the order is relevant for translation. Regardless of conjunction used the order is always important for translation. In fact a few people on this thread have queried why translating the adjectives in the opposite order is not accepted - because the order they are translated in should remain the same. The Japanese doesn't say quiet and pretty so the English shouldn't either.
きれい can mean either pretty/beautiful or clean/tidy. I'd venture to say neither translation fully encapsulates the word in Japanese, but they are the closest translations based on context. I THINK that clean would probably be the more likely interpretation here, and you'd use a more explicit term if you were trying to say the park was pretty.
と is used as and only with nouns eg. パン と ケーキ が ほしい です - I want bread and cake. で or the te form of verbs is used as and to join sentences eg. (using the original sentence as an example) the two sentences being joined are その 公園は きれい です - that park is pretty/clean and しずか です - it is quiet. The te form of です is で. By changing です to で the two sentences can be combined その こうえんは きれい で しずか です - that park is pretty and quiet. Here is another example - としょかん に 行って べんきょう します - I go to the library and study - 行きます is changed to its te form 行って so that としょかん に いきます - I go to the library and べんきょう します - I study can be combined into one sentence.