Why there is no 的 after 你 in 你国家的医院好吗?
This question is really about: &quot; when can we omit the possessive 的 &quot;
的 1.(adjective suffix):
For example:

强大勇敢 (strong and brave)

强大勇敢的人 (strong and brave man)


You cannot omit the adjective suffix 的 and write 强大勇敢人

高大 (tall and big)

高大的人 (tall and big man)


You cannot omit the adjective suffix 的 and write 高大人
Adjectives that do not need adjective suffix can directly apply to the object. 

For example 好(good) in 好女儿(good daughter)

的 2.(possessive):
我(的)衣服的鈕扣的顏色是白色的。 can be reduced to: 我衣服鈕扣的顏色是白色的。
(The last 的 is not omitted because it is not a possessive 的 but an adjective suffix 的.)
The same with your example sentences:

你(的)国家的医院好吗？can be reduced to: 你国家的医院好吗?

我(的)妈妈的房子很高。can be reduced to: 我妈妈的房子很高。


From user3a's answer:

The “的” before the subject is usually kept.
Others can be omitted, if possible.

I say it is wrong. The possessive '的' after a subject is usually kept, if it is a 'true possessive'(something you actually own) For example:  The possessive 的 in 我的車 cannot be omitted and make it &quot;我車&quot;
On the other hand, if the '的' is a 'relative possessive'(something you don't actually own) can be omitted. For example: you can omit the '的' in '我的母亲' and write 我母亲.
More example:
The possessive 的 in 这是我的太太; 他是我的女儿; 你的国家; 你的公司 can be omitted. You can write 这是我太太; 他是我女儿; 你国家; 你公司 (if you actually own the country or the company, then the possessive 的 should be kept)
Again with your example sentence:

我(的)妈妈的房子很高。 can be reduced to: 我妈妈的房子很高。

我(的)牛(的)角很长。 cannot be reduced to: 我牛的角很长。 or 我的牛角很长。


(you possess the cow and the cow possesses the horns, neither can be omitted.)