第 [dai-] vs 目 [-me] in forming ordinal numbers

Question

As I understand fo now 第 [dai-] and 目 [-me] both turn 5 to fifth.

五匹の犬 [go-hiki no inu] five dogs
五匹目の犬 [go-hiki-me no inu] the fifth dog

Here everything is clear.
Articles and books say you have to use counters in Japanese.
匹 is counter for small and middle-sized animals.
Manuals say, that you can not use numbers without counters

五目犬 [gome inu] the fith dog - illegal
五犬目 [go inu me] the fith dog - illegal

They say, that you must use some counter here like this.

五匹目の犬 [go-hiki-me no inu] the fith dog - legal old school, classics

Ok, everything is clear to me.

But here comes 第 [dai]...

• (Opt-1) 第五犬 [dai go inu] the fifth dog
• (Opt-2) 第五の犬 [dai go no inu] the fifth dog
• (Opt-3) 第五匹の犬 [dai go-hiki no inu] the fifth dog
• (Opt-4) 第五匹目の犬 [dai go-hiki-me no inu] the fifth dog

OMG!!! WHAT THE HELL?!

My questions:

• Q1: Which of these 4 options are legal?
• Q2: Does all 4 have sence?
• Q3: If yes, if there any difference between them?
• Q4: If Opt-1 is legal and ok why it is allowed to use numbers without counter?! In every article on counting there is info, that you always have to use counters. But what, if you use [dai] no counters for [inu] is allowed?!

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第 [dai-] as [The fifth dog] vs [The dog #5]

I know, that 第五犬 have additional meaning for [Dog #5]
It is equvalent

• to 第5犬
• and to 五番の犬
• and to 5番の犬

As I understand all these "number"-forms does not relate to the "fifth dog"-meaning.
Therefore my questions are about [the fifth dog], not the [dog #5].

• Q5: What options in Japanese to say "the fifth dog #5"?

• (Opt-F5-1) 第五第五犬 [dai go dai go inu] the fifth dog #5
• (Opt-F5-2) 五匹目の第五犬 [go-hiki-me no dai go inu] the fifth dog #5

The meaning of the imaginary situation:
there are 5 dogs, each of them has #5 on the badge
we see the first dog #5
we see the second dog #5
...
and finally we see the fifth dog #5.

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A screenshot from new anime "OVA Re: zero "Memory Snow" which shows example of using dai with counter but for some reason without の [no].
I thougt that の [no] MUST ALWAYS go after counter, but here we see real example without の [no].
Strange thing....

Answer 1

As far as I know, 目 is used to talk about the position of something that is not structured or categorized beforehand and whose elements are equal if you disregard their position. On the other hand, 第 is used with elements of a set that conform a fixed, logical structure made of units, where each of those units has a particular meaning or content that makes it different to the other ones, regardless of its position.

For example, to enumerate the cars running in a race, you would use 目, as you can't know how they will be ordered beforehand. 一つ目の車. You would also use 目 when enumerating shelves of a closet （上から ）一段目の棚、二段目の棚… They can be counted either from the top to the bottom or from the bottom to the top, and appart from their relative position to each other, there's no difference between them.

As regards 第、some examples might be chapters of a ドラマ, lessons in a textbook, and so on. They are organized in units that conform something larger. Chapters are units of a season in a tv show, lessons are units of a textbook, etc.

Summarizing,

X目の○ → the X th ○ .

and

第X○ → the ○ # (number) X

Disclaimer: this answer is based on my own, limited experience so don't assume what I'm saying here is right. Hope it helps!

Answer 2

Just 2 answers in reversed chronological order:

First, about the screenshot, there is no の　since the 第一回　does not refer to the snow festival (ie it is not about "1st snow festival"), but in this case means "1st episode"

About Q5: There could be other acceptable ones as well, but I think that 五匹目の第五番の犬 would be best. But if the tags on the dogs only had a "5" written on them (no 第), I would say 五匹目の五番の犬

I need to stop here, due to both lack of skill and since "Daigo no inu" just makes me think of a Japanese TV show where a famous タレント by name Daigo is taking care of dogs with fears and other behavioural problems resulting form their traumatic past.