First of all, for your question, between 6&7, 7 is right! 我去看了病
Let me put it this way: the former example you mentioned 我去了中国 and 我去中国了 are both right; but between 我去了看病 and 我去看了病, only the 我去看了病 is correct. The difference lies on the word right after 了 here, 看 from 看病 is a verb, you cant connect two different verbs with 了 in the middle(except: 看了看/听了听/试了试 ect,, the first verb and the second verb are the same); however, you can put a noun right after 了, in this case, '我去了中国'---中国 is a noun.
more examples for your understanding:
verb+了+noun: (correct)
丢了钥匙。
我看了电影。
我做了作业。
我去了电影院。
verb1+了+verb2 (wrong)
我去了看电影。
我去了吃饭。
我去了上课。
Note1: you will notice my examples are all with 去 bla bla, this is because verbal phrases are often formed with this pattern 'go to do something ' in English too(go to do homework, go to travel,etc.)
Note2: if you want to correct, there are two ways, one way is just omit '了‘ (我去看电影;我去吃饭)or put 了 behind the sentence, 我去看电影了;我去吃饭了。 The two patterns are correct grammatically, but have different emphasis.
If i have time, i would love to talk about this more, but i gotta go to have a lesson now!
Hope this makes sense to you!
Good luck!