Correct me if I'm wrong, but I feel like this question arose from a misunderstanding of tenses in the example sentence

昨日起きるところで (Link example)

from your link's answer.

The key is noticing that 起きる is a relative clause characterising ところで. Thus, this tense is unrelated with the overall sentence tense (which since 昨日 was used, has to be past).

So, do your patterns exist? Well, they do appear together but not in the way you're probably thinking. Let's see how.

1. (Changed from 食べる to 買う since I couldn't find a meaningful example with the former) 明日買った makes no sense on its own.
   However, if I put it like 明日買った本を読みます it is perfectly fine. Why? Because 明日 is in accordance with 読みます while 買った is a relative clause characterising 本.
   Overall, I'm saying that "I'll read the book, which I bought yesterday, tomorrow."

2. (I'm assuming you meant 昨日, not 今日. Also, changed the verb for the same reason as above) Again, if I use 昨日出かける on its own it makes no sense.
   But in 昨日出かける時、鍵を忘れてしまいました all is fine. 出かける is characterising 時, and 忘れてしまいました is in accordance with 昨日 so there is no incompatibility in the overall sentence.
   Overall, I'm saying that "When I left yesterday, I forgot my keys".
   Notice how there is no direct translation here. In English, you use "left", but in Japanese you use "出かける" since leaving takes place after forgetting the keys.

3. Couldn't actually figure out what you meant with this one =(

Finally, regarding I and II, let me start by telling you that II is actually incorrect. According to the Genki textbook, when giving advice in the form ～ほうがいいですよ(normally written in かな) the tense of the advice is a bit peculiar.

When giving affirmative advice, like 早く寝る, the verb must come in the past short form. So, 早く寝たほうがいいですよ is the correct spelling.

Conversely, when advice are in the negative, negative short form is used instead. In your case, this would be 早く寝ないほうがいいですよ。

Hope this helps ^^

Edit:

Regarding the 昨日出かける時、鍵を忘れてしまいました sentence. Originally, I intended to include an explanation of the tense usage, but was a bit long and not entirely relevant. In any case, detailed information can be found in the Genki II textbook.

To summarise it briefly, in a sentence of the form A 時、B, the tense in A indicates the time at which A took place relative to B. Here are two examples from the book.

チベットに行く時、ビザを取ります。
I will get the visa issued when I go to Tibet.

When the action in A occurs after the action in B, A is in the dictionary form. In this case, you first take the visa and then go to Tibet.

中国に行った時、ウーロン茶を買います。
I will buy oolong tea when I go to China.

Conversely, you use A in the short past when A happens before B. In this case, you first go to China, and then buy the tea.

Note that in both cases, the "overall" tense is given by the verb at the end, which both indicate future in this case. Both sentences can be put in the past (if you've already gone to China/Tibet) by simply changing 取ります into 取りました and 買います into 買いました. However, the relative order between events remains the same, so the tense in A is unchanged.

Regarding the advice, let's break it down fundamentally. First, you identify the particles: in this case, が marks the subject and よ merely adds emphasis at the end. Now, you need a noun before が, in this case ほう. 早く寝た is just a relative clause characterising ほう.

Generally speaking, relative clauses can be in any tense. Japanese writing usually employs really long clauses just to qualify a noun. Simple examples follow:

これは私が買った本です。
This is a book that I bought.

眼鏡を掛けている人があそこにいます。
There is a person wearing glasses over there.

So, to put it bluntly, 寝た and いいです are not in the same clause. 寝た makes part of a relative clause that adds extra information, but is not necessary. Of course, if I just say ほうがいいですよ this is correct, but you'll probably ask me 何がほうがいいですか？ To which I would respond, 早く寝たほうです.

As to why you only use the short past for affirmative advice with ほうがいいですよ, I'm afraid I don't have an explanation. Sometimes, it's just how things are.