Is it possible to estimate the birthday of a language based on vocabulary size (German language has probably larger vocabulary size than English language)? For example, Germanic languages contain a set of languages, among which is the English language, which later was inspired/affected by the French language. How we can know if English is older or younger than German if that is reasonable question to ask in the first place for linguistics?
Since you asked, it is not a reasonable question to ask in the first place. Multicellular organisms have birthdays, languages do not. Every existing German speaker learned the language from surrounding German speakers, who themselves learned German from other older surrounding German speakers and so on. There is no crisp distinction between the spoken languages corresponding to "Contemporary Modern High German", "Early Modern High German", "Late Middle High German", "Early Middle High German", "Old High German" and so on. At some point, the ancestor of English was the same language as the ancestor of German. The idea of "German" as a unified language, identifiable as distinct from some other language (Dutch? Old Dutch? Old Saxon? Old English? Langobardic? Old Frisian?) may be based on invalid non-linguistic premises ("People living in modern Germany speak German, people living in modern England speak English"), or they may be based on actual linguistic criteria – the changes in language that characterize the languages that end up being known as English, or German.
English and German are both West Germanic languages. One can linguistically study the properties of the proto-language "West Germanic", and understand what linguistic changes characterize the development of the language that is the antecedent of English, and the antecedent of German. The problem is, as the tree diagram of West Germanic here indicates, German comes from two different branches of Germanic and the development of "German" as a unified language as a single language is more recent.
In short, the problem is that we don't know when a language is "born", in fact it makes no sense to talk of a distinct birth of a language. Language is more like a bacterium: it splits, and splits again, ad infinitum.
Just as an addendum, the Khoisan language Shua is spoken about 150 miles from where the Bantu language Shona is spoken. Shua and its ancestors has been spoken in that are for perhaps tens of thousands of years, Shona came into the area maybe 1500 years ago. Shua has about 10,000 words, Shona has about 10²⁰ words. The reason for the vast difference in numbers of words is that Shona allows words to contain many distinct meaningful units (tense; negation; reflexive; diminutive...) and Shua mostly uses a sequence of separate words to express the same ideas. The "number of words" in a language is mostly an accident of grammar and not the result of how long a language has been spoken in some place.
Since you're asking specifically about the "genesis of the language":
For most languages, the "genesis" is unknown and probably unknowable. I speak English because I learned it from someone who learned it from someone who learned it from someone…all the way back about six to seven thousand years. That's how far back our current methods let us reconstruct it; we can come up with surprisingly good guesses for how the ancestors of English would have been spoken up to about 4500 BCE.
Before that…we just don't know. Surely the people who spoke an ancestor of English six thousand years ago also learned it from someone, who learned it from someone, and so on, but the comparative method (which lets us reconstruct back that far) just can't give reliable predictions any farther than that.
If you go back far enough, there are various different theories for how language originated. Some people think that the properties we see in all modern languages (like recursive syntax) were invented several different times by different groups of people, once we had a brain that was equipped to handle it. Others think the jump from a communication system to a complete "language" happened only once and then spread from there. Or maybe it was punishment for the hubris of humanity trying to build a really big tower. We just don't know.
Now, there are some languages where we can trace their origin. These are creoles. When people who don't share a common language are put in contact, they'll often improvise a rudimentary communication system called a pidgin; when children grow up using this pidgin, it can evolve into a more complex, full-fledged language called a creole. (Note that many languages named some variant of "pidgin", like Tok Pisin, are actually creoles by this definition.)
For creoles, then, we can actually say where and when the language was born. Haitian Creole, for example, originated in the 18th century. But for most languages, a definite start point like this isn't possible to determine.
P.S. So if we can't actually pinpoint a starting point for English, you might ask, how do linguists talk about different stages, like "Proto-Germanic" versus "Old English" versus "Modern English"?
The answer is, these names are a convenient shorthand rather than a quantitative reality. There was no sharp dividing line when people stopped speaking Middle English and started speaking Early Modern English instead; it was a gradual process of evolution, and scholars somewhat arbitrarily picked a point in the middle to make it easier to talk about the different stages.