Old grammar forms live on in American English

Yes, but what it consists of is a perennial source of controversy. :-)

https://en.wikipedia.org/wiki/Linguistic_universal

On the one hand, there are lots of concepts that are found in lots of different languages (so understanding a concept in one language is a big help for another language). Noun case is a great example of that, but it's use to different extents in different languages, and languages that have noun case don't always use it as extensively as one another (for example, in Latin it's arguably marked directly on every noun and adjective, even on names, while in English we nowadays only have it in pronouns!).

Also, languages have different numbers of cases (some people say that English just has subject and oblique, though perhaps also a genitive, while German still distinguishes a dative, Latin also has an ablative, and Finnish and Estonian famously have dozens of cases).

You definitely can't use noun cases as a linguistic universal, though: for example, Chinese doesn't have them at all. There Barney's "I love you, you love me" should just be "我爱你，你爱我" (I love you, you love I). Whereas in Latin it would most idiomatically¹ be "ego te amo, tu me amas"—every single word is different between the "I love you" and "you love me" cases, and the grammatical relationship is still also reflected (although not explicitly expressed²) by the word order.

So it's one thing to benefit from learning grammatical concepts that you can apply to new languages as you learn them (as well as to your native language), and another thing to expect to encounter each concept in each new language you look at, which is most likely not going to happen.

There are also two different phenomena which are likely to make some grammatical features feel recognizable:

https://en.wikipedia.org/wiki/Language_family

(the fact that many languages are historically related to one another) and

https://en.wikipedia.org/wiki/Areal_feature

(the way that languages that are unrelated to one another by descent nonetheless influence each other through language contact within a geographic area, or sometimes trade or religious influences)

Many of the languages that you encounter in Europe, in former European colonies, and in Central and South Asia (although not Hungarian!) are part of the Indo-European family.

https://en.wikipedia.org/wiki/Indo-European_languages

These languages share a common descent (and include all of the Romance and Germanic languages, and many, many others). They often have features like prepositions, participles, noun inflections (like case markers), verb inflections (perhaps tense, number, mood, voice, and person, although not all languages include all of these distinctions), and agreement (certain forms of words have to match certain forms of other words in the same sentence in order to confirm a grammatical relationship).

The Indo-European influences as well as areal effects have also resulted in a phenomenon sometimes called "Standard Average European"

https://en.wikipedia.org/wiki/Standard_Average_European

where European languages are, by worldwide standards, grammatically rather similar to one another. But these similarities shouldn't all be taken to be linguistic universals that reflect how the human mind works, because many other languages don't have them, and may also have other features that are rare in European languages. (Some examples that are somewhat common in languages overall but rare in European languages and in the Indo-European family include ergativity and evidential markers, while marking number on nouns or tense on verbs is less common in languages from some other parts of the world.)

¹ This example uses subject pronouns, even though they are optional in Latin (unlike English). https://en.wikipedia.org/wiki/Pro-drop_language

² This example uses SOV order, which is the most common order in Latin but isn't grammatically required at all due to the noun case markings. For example, "te amo ego" is less idiomatic but is still grammatically acceptable and still has the same meaning.

I have my doubts; I still cling to Whorfian ideas of linguistic relativity and suspect the governance and binding framework doesn't explain more nuanced or culture-specific aspects of syntax like comparisons or kinship systems, but it's hard for me to argue with the success they've had in schematizing a ton of language families under a simple-ish model.

Disclosure: not a linguist, but I play at one.

That's more a result of your education than your native language. (And my education, until one German teacher was frustrated and ran through English grammar, so she could refer to it for teaching German grammar.)

English teaching in Britain and the USA has periods of not teaching English grammar. French students in France study French grammar, Spanish students in Spain study Spanish grammar.

Friends working in academia say they can recommend to their PhD students, "I think you should rephrase this in the subjunctive", and it's almost always only the British and American students that don't understand.

The first kind are any word-creation suffixes that change the meaning of the root word, perhaps making it a different part of speech. Such a suffix is called a 'képző'.

The second kind are word-relationship markers that preserve the part of speech but signify a clarifying quality, like belonging to an adjacent noun, or plurality, comparative intensity, or an ordinal pointer -- the sorts of markers you'd use to answer questions like 'whose?', 'which?'. This kind of suffix is called 'jel'.

The third are role-in-sentence markers that signal the function of the word within the sentence. This suffix is called a 'rag'. For verbs, these are endings for grammatical person and mood. For nouns, the most common of these correspond to nominative case, accusative case, and dative case, but then there's one that expresses the notion of being used as an instrument, one that expresses a purpose, two that are used to signify a transformation, and a couple others. And, there are 10 more that show a location relationship; English would prefix such words with one of: at, into, in, out of, towards, by, away from, onto, on, from off of; in Hungarian, the word takes a corresponding suffix instead.

When I was learning English grammar in an American school, a structural examination at a level I was used to from Hungarian was largely absent -- it took until I was learning English as a foreign language, or learning German as a foreign language, that mechanics of English grammar became apparent to me. The dying usage of subjunctive mood and incorrect use of accusative and dative case is a natural simplification, but enabled by the lack of education about the origin of those forms and their meaning.