Yulia, I’d like to try to answer your questions in a way that will be useful to you.  Please advise if I have misunderstood what you’ve asked, which I believe to be this:

1. Can it ever be grammatical to say *“six **feet** five”* instead of the normal *“six **foot** five”*?<br><br>
 **Answer**: Not really; if you say *feet* there you’d best add *inches* afterwards. Leaving it out doesn’t sound right to me anymore.

2.  Does including *inches* at the end of *“six foot/feet five **inches***” affect the grammatical number of *foot* to make it *feet*?<br><br>
 **Answer**: Yes, it sometimes can. See below.

I had hoped that our site’s earlier questions regarding the grammatical number we use with measuring units would answer your question. Candidates include:

* [Why don't we pluralize “foot” in measurements?](https://english.stackexchange.com/q/28457/2085)
* [Pluralization rule for “five-year-old children”, “20 pound note”, “10 mile run”](https://english.stackexchange.com/q/1366/2085)
* [Height and Weight — How to write them when abbreviations are not used](https://english.stackexchange.com/q/154230/2085)
* [Can “zero hours contracts” be considered as not grammatical?](https://english.stackexchange.com/q/322241/2085)
* [“40-50 years old” vs. “40-50 year olds” when referring to a group](https://english.stackexchange.com/q/88205/2085)
* [Does one hyphenate height when given in feet and inches?](https://english.stackexchange.com/a/122918/2085)

I could not find a question or answer that exactly matched your situation. The general rule to remember is this:

>### In English, the grammatical number of a numerically quantified measuring unit is not inflected when used *attributively* (immediately before a noun).

This is not some special rule that applies only to measuring units alone. It is a general rule that applies to all combinations of counting numbers and the nouns those numbers are counting.

Here are examples of numbers and their noun used attributively before another, so those nouns must be in the singular even if the number is other than one:

1. A four-door car would not have that problem.
1. John and Jane and James were in a three-person marriage.
1. Even a thousand-mile journey begins with its first step.
1. The only thirty-day months are September — and April, June, and November.
1. Metric-rights activist Monsieur La Douleur comically insisted that seven-league boots had become illegal “après la Révolution”.
1. My three-week four-day vacation was just one day shy of a full month.
1. Our new seven-pound, three-ounce baby came into the world at three o’clock this morning.

In contrast, here are those same examples recast so that the number and noun are no longer used attributively, which means now that number governs the grammatical number of its noun immediately following:

1. A car with four doors would not have that problem.
1. John and Jane and James were in a marriage of three people.
1. Even a journey of a thousand miles begins with its first step.
1. The only months with thirty days are September — and April, June, and November.
1. Metric-rights activist Monsieur La Douleur comically insisted that boots covering seven leagues in a single step had become illegal “après la Révolution”.
1. My vacation of three weeks and four days was just one day shy of a full month.
1. Our new baby came into the world at three o’clock this morning weighing seven pounds, three ounces.

The last two examples in each set deliberately involve more than one unit, just as your own case of six feet and five inches (6ʹ 5ʺ) does.  Let’s look at this set:

8. John Clease is six foot five. (*this is actually the weird one*)
8. John Clease is six *feet five.  (*not grammatical <sup>for me but YYMV</sup>*)
11. John Clease is six feet, five *inch.  (*not grammatical*)
11. John Clease is six feet, five inches.
11. John Clease is six ?foot, five inches.  (*may not be grammatical for all native speakers*)

11. Six-foot-five actor John Cleese towers over his cohorts.
11. At six feet five inches, actor John Cleese towers over his cohorts.
11. Towering over his cohorts, actor John Clease stands a full six foot five. (*another weird one<sup>(?)</sup>*)

If you say *inches* in the plural, one would expect *feet* in the plural as well. But one can find examples where this is not always the case, where instead singular *foot* has been used with plural *inches*. Whether this is grammatical for all native speakers is unclear.

Not all these John Cleese examples follow the patterns we’ve seen before, and it isn’t clear why they differ. For example the predicate use can remain uninflected, while before all predicate examples inflected for number. That’s not true for babies:

16. My baby is seven pounds three.
16. My baby is seven *pound three. (*not grammatical*)
16. My baby is seven pounds, three ounces.
17. My baby is seven *pound, three ounce. (*not grammatical*)
17. My baby is seven *pound, three ounces. (*not grammatical*)

So you would in theory think feet and inches would always work exactly the way pounds and ounces work, but in practice, they do not.

The bottom line is that if *six **feet** five* is never grammatical used attributively before the noun. It can only be:

13. Six-**foot**-five actor John Cleese towers over his cohorts.

On the other hand, *six **feet** five* used predicatively is only grammatical when it is immediately followed by *inches*: 

11. John Clease is six feet five inches.

I cannot tell you “why” it works this way, nor why it doesn’t work some other way. I can only report what does or does not sound right to a native speaker.