Can you only use 又 with adjectives, or can you also use it with verbs, verb phrases, and nouns？ I want to say, "Summer this year is rainy and cloudy.", so I wrote, “今年夏天是又雨天又云天。” Is this correct, or should it be something else, such as “今年夏天又是雨天又是云天。”?
"Summer this year is rainy and cloudy."
Your English sentence denotes that "rainy and cloudy" are the attributes in this summer, implying the weather in a given day of this summer could either be rainy or cloudy.
So, if that's what you want to express, then you can use 要么...要么.... 要么...要么... expresses a logic OR, while 又...又... expresses an logic AND. You can put:
You can also interpret "rainy and cloudy" (as a whole) to 阴雨天. Rainy is 雨天 and cloudy is 阴天. Combine together: 阴雨天. 阴雨天 means that the day is cloudy and it can possibly rain at anytime. If you say this summer is all 阴雨天, it can mean that cloudy or rainy had gone throughout this whole summer. If this is what you meant, you can put:
今年夏天都是阴雨天 or 今年夏天尽是阴雨天
Your Chinese sentence "今年夏天又是雨天又是云天。" is grammatically correct, but it's logically incorrect. If you said it in practice, I would have asked 那今年夏天到底是阴天还是雨天呢？ （whether this summer are rainy or cloudy）.
Hope this can help.
This webpage provides good reference又……又 usage
However, it is inappropriate to use 又……又 here. Because 又……又 indicates simultaneousness but I think the weather cannot be both cloudy and rainy. You may want to express that it's always rainy or cloudy this summer, then I recommend “今年夏天多是阴天和雨天” or more literary “今夏云多雨绵”