Why is 的 used in 你是哪一天出生的 and 我们什么时候开始的 and 前天你什么时候来的这里?
的 at the end of sentences can be seen in really a lot of cases. Here are just two cases I can think of at the moment:

case 1. 你(是)哪一天出生的？

This question asks about an event which happened in the past. Note it often comes with 是.
If we omit both 是 and 的, it will become 你哪一天出生？This sounds a little weird, because generally the new version means an event to happen in the future, but in the context people understand this is asking about an event in the past.
Compare:

A. 他(是)哪天来的？- asking about an event in the past.
B. 他哪天来？- asking about an event to happen in the future.

Another example:

他在20年前出生。

There isn't 是 or 的, but people understand it without a problem and this is used a lot. It's also very common to see 他(是)(在)20年前出生的。

case 2. 树叶是绿的。 - The leaves are green.

This sentence describes the current status, or the general case.
Generally people don't say 树叶绿 as a standalone sentence, because it is incomplete. However this fragment can be used in poems or nursery rhymes(儿歌，童谣), for example:

黄瓜绿，番茄红，辣椒秧上挂灯笼。

If people say 树叶绿了, this means &quot;the leaves turned green&quot; or &quot;the leaves have turned green&quot;.
Note: in Chinese, there are a lot phrasal patterns, like 在 ... 以后, 从 ... 中, 虽然 ... 但是, 不是 ... 而是, and a lot others.
是...的 is one of the most used phrasal patterns. Its structure is

Subject + 是...的 (是 sometimes can be omitted)

This structure can be used to give general descriptions, or to describe events which happened in the past, or to express possibility of something to happen in the future.
Examples:
他(是)前天来的。- describe past event
我(是)坐车来的。- describe past event
我们(是)一起来的。- describe past event
我(是)喝着草原的羊奶长大的。- describe past event
我的一个朋友是被他的女友杀害的。- describe past event. 是 not omitted before 被
我(是)不会弃我的目标的。- express willingness
我看，他(是)不会来的。- express possibility of future event

Although we can sometimes omit 是 from the 是...的 pattern, but we don't ommit 的。We can say, 树叶是绿色的。But we can't say 树叶是绿色。 Because 绿色 is the name of the color (green), and 树叶 is not a type of color. Just like in English, we say &quot;the leaves are green&quot;, where green is an adj. But we don't say &quot;the leaves are green color&quot;, while we can say &quot;the leaves are of green color&quot;. Here the word &quot;of&quot; servers a very similar function as 的 in 树叶是绿色的。
The 是-omitted variation is used in poems a lot. In the following examples, 是 is omitted:
树叶绿绿的。
她的脸红红的。
时间的脚步轻轻的，走过了我的窗前，也走过了我的心田；走过了我的留恋，走过了我的思念。