I know that there was a large influence on English by norman french and to a certain extent by latin itself on vocabulary and pronounciation. But did it also change grammar, syntax etc. ? For example if you take the word 'go' (or 'to go'), the past tense of it is 'went', making it an irregular verb and basically replacing 'go' with a completely new word in the past tense. That is very unlike most other germanic languages (eg in german 'gehen' -> 'ging')
So did this come from the latin 'venit' which does roughly mean the same? Especially since pronounciation of classical latin 'v' is kind of similar to the english 'w', at least at the beginnig of words.
The impact on pronunciation was actually fairly minor. The biggest impact by far of Norman French on English was lexical (vocabulary), as you said, with a smaller but substantial morphological effect (especially prefixes and suffixes; see below).
For our purposes, Latin played essentially no direct role. Latin is the background of French, however, and so Latin -> Norman French -> Middle English. Many later English words were generated based on Latin archetypes, but most of these are via French anyway. You have noticed Lat. venit with English "went," but this is pure coincidence, and I should note that Latin venire means "to come," not "to go." They seem very similar but they are worlds apart as far as their origins. English "went" is grafted onto the paradigm "to go" from the verb "wend," as we see in the Prologue of Canterbury Tales:
And specially, from every shires ende
Of Engelond, to Caunterbury they wende...
Germanic languages, including English, and Latin share the same mother, Proto-Indo-European, but the roots for to come, to go, and to wend are all complete different. The PIE root for "to wind" is **(H)uendh-e/o-*, giving us Gothic -windan, Old English windan, New High German winden.
As far as syntax (or grammar, as you say), the impact of Norman French is also small, but not completely absent. Impersonal constructions in English probably stem from, or were encouraged by, French (e.g. "one does not simply walk into Mordor"). Adjectives in the attributive position (ie after their noun, like Poet Laureate) are thanks to French influence. The standard -s ending for pluralization in English (cat, cats) is probably due to French, in part at least, though Old English had -as for strong masculine nominative and accusative plurals and this certainly played a part. I would count that one as a joint effort. Probably most substantively, English love of gerunds ("I am ok with running but I die for swimming") is was no doubt strongly encouraged by French gérondif constructions (again, via Latin). This is a standard list and I'm probably leaving a few out, but that's a good survey I think.
Many don't realize the fundamental impact of French on word-building in English, however. French morphemes, via Latin, play a huge role in word generation and preference in Middle English. Of the suffixes, -tion (nation, ration, vacation, subjugation) is extremely productive, mostly from Latin or on analogy (randomisation, standardisation, with American English preferring the -z-). Also busy is -ous "full of" (e.g. wonderous, "full of wonder"), directly from Latin suffix -osus/a/um "full of;" compare this to the native Germanic suffix -ful, like "wrathful" or "hopeful"). The suffix -ment (again Latin --> French) is common (establishment, cement, pediment). The suffix -able (from Latin -bilis) also deserves special mention and is very versatile in English (hackable). Of the prefixes via French (again from Latin), I will only mention pre-, which again is very common (preview, prefix).
Many native Germanic suffixes of course also hang around in English: guileless (a Germanic sufffix on an Old French root), perfectly (a Germanic suffix on a Latin participial root), hotness (Proto-Germanic haita-), "your lordship" (Proto-Germanic -skepi), and so on.