Both 还 and 又 may be used to indicate an addition to the preceding information. For example:
May I exchange both in all above examples? If not, what are the rules involved?
Short answer according to what I could understand from Frank Joe's reply:
The difference between 还 and 又 is that 又 expresses a feeling like "Again?".
For example, "I ate spaghetti yesterday and today I ate it again!" You can use "我昨天吃了意面，今天又吃了意面!"
In this sentence, 又 means that you want to emphasize that you ate it 'again'. Maybe you want to express that spaghetti is your favorite, or you want to express that you have nothing left in the refrigerator so you just have to eat it again. When you use 又, you always want to let the other person focus that the action is repeated.
But 还 just means "also". There is no special meaning here. In your example, 我买了香蕉、苹果还买了葡萄, you just want to say that you bought banana, apple and grape. There is no relationship between these three fruits and you don't have other meaning for 还.
More specific for your question, you cannot exchange 还 and 又 in your example.
昨天没考好，现在又撞伤了，真倒霉。
This sentence uses 又 because the fact that the person didn't do well in this exam is already bad news. What's more, he is injured now, which makes this person more unfortunate. You should find that both 没考好 and 撞伤 are negative things. In this sentence, bad things are repeated. We use 又 here because we want to emphasize how unlucky he is in this period of time. 又 deepens the level of unluckiness of this person.
他要买肉又要买鱼。 The speaker tries to emphasize that he needs to buy both meat and fish. You didn't specify the background of this sentence. But if he wants to do this buying, you should use 还 here. If he is not willing to buy these, 又 is more appropriate.
我买了香蕉、苹果还买了葡萄。 I just bought these three fruits. The speaker wants to state this fact and nothing else.
我们不但要努力学习，还要考得好。 For this sentence, the attitude is positive. We need to study hard and get perfect on exams and we really want to reach these goals. If you want to use 又 here, the background should be "our parents ask us to do these and we think these requirements are impossible to do". And you should replace 不但 with 既 because 既...又 is a word group.
All in all, the usage of 又 and 还 really depends on the background of specific sentence.
都是副词，作状语，以下情况时经常可以互换：
用于未发生的行为，表示同一行为的重复，常与＂要、想、得、会＂等助动词连用，动词后面没有＂了＂：
１）假如你下次还来找麻烦，我就不客气了。（又✓）
２）下次还遇到这种事，我该怎么办？（又✓）
３）他上个星期去了白云山，这个星期还想去。（又✓）
４）我昨天去银行取钱，没取到，今天又得去一趟。（还✓）
５）今天听写，明天又要听写。（还✓）
６）如果不制止，他以后又会这么做的。（还✓）用于过去的行为，表示不同行为的增加，动词后面有＂了＂：
１）昨天我买了一支笔，还买了几个本子。（又✓）
２）小明刚才扫了地，还擦了窗户。（又✓）
３）下午我去邮局寄了信，又去市场买了一点菜。（还✓）
４）去年小周买了房，又买了车。（还✓）
用于未发生的行为，表示行为重复时，如果句未有＂了＂，或前面有否定词，只能用＂又＂，不能用＂还＂：
１）要是这次又失败了，那可怎么办？（还✗)
２）如果你今天又迟到了，就只能坐在最后一排。（还✗)
３）你千万不要又和别人吵架啊！（还✗)
４）你可别又发脾气啊！（还✗)＂又＂还可以用于已经发生的行为，表示同一行为重复，动词后或句未常有＂了＂；＂还＂不能这么用：
１）阿里昨晚又喝醉了。（还✗)
２）他把信看了一遍，又看了一遍。（还✗)
３）他又来找麻烦了！（还✗)
４）你怎么又惹妈妈生气了？（还✗)＂又＂还可以用于有规律性的客观行为、事件，表示重复；＂还＂不能：
１）冬天到了，天气又变冷了。（还✗)
２）新的一年又到了。（还✗)
３）下星期又该考试了。（还✗)表示增加意义时，＂又＂可以表示不同时间范围内行为的增加，＂还＂不能：
１）去年他买了房，今年又买了车。（还✗)
２）我前年去了欧洲旅游，今年又去了东南亚几个国家。（还✗)都可以表示行为或状态的持续，但不能互换：＂还＂修饰的动词后面不能有＂了＂，大致相当于＂仍然＂；＂又＂修饰的动词之后要有＂了＂，大致相当于＂接着＂：
１）都１０点了，你怎么还睡？(又✗)
２）那一年，我大学还没毕业。(又✗)
３）都几天了，她还在生气呢！(又✗)
４）你走后，我们又聊了十多分钟。（还✗)
５）叶明又休息了几天，才来上班。（还✗)
６）时间还早，我又睡了一会儿才起来。（还✗)
One major difference is when using 不但/不仅....还, you shouldn't substitute with 又 in most cases, but you can in certain scenarios. 她不仅会唱歌，又会跳舞，而且还会画画.
Your second sentence sounds a bit awkward, either the logic is not there or not how I would say it. Either 我们只有努力学习，才能考得好。 Or 我们不但要努力学习，还要保持/注意身体健康. The latter could easily be a slogan in a typical Chinese school.
