The two largest sources of English vocabulary are about equal in size:
1. Germanic (mostly Anglo-Saxon, aka Old English, and Old Norse). Note that many people misinterpret Germanic to mean "from German". German, English, Danish, Norwegian, Swedish, Dutch, Afrikaans, Icelandic, Faroese, Frisian, etc all come from a Proto-Germanic. They are all cousin languages.
2. Old French (which evolved from Latin), equal in size to #1.
3. The next largest source is Latin itself.
4. Followed by ancient Greek, often through Latin (which adopted a ton of Greek words).
5. The remaining 10% comes from dozens of other languages. No single one of them contributes a significant amount of vocabulary by itself.
So, Old French and Germanic sources combined are about equal. However, if you include Latin itself with Old French, then "Latin languages" do outnumber Germanic sources.
Other Latin languages and other Germanic languages are in the 10% and do not contribute a significant number of words. Of course, many words among the Germanic language and many among the Latin ones often look similar, but it's mostly not from direct borrowing. The words mostly come from Old French or Latin directly, or from Old English or Old Norse directly.
The answer to your questions very much depends on the wording and definitions. If Latin languages means only Latin evolved languages, then NO, it's not true. Germanic sources and Old French are about equal. If Latin itself is included, then YES. If you truly mean other Latin evolved languages in addition to French, then not really. There are several words from other Latin languages, but not a significant amount.
English grammar is mostly simplified Germanic. Most Germanic languages have simplified their grammars from Proto-Germanic, but to varying degrees. English and Afrikaans have simplified the most.
Grammatical influences from French are minimal. It probably is responsible for English strongly favoring SVO (subject-verb-object) word order, instead of V2 (the verb must be the second element, meaning that both SVO and OVS occur, in main clauses) and instead of SOV in dependent clauses, like most other Germanic languages.
However, English favors Question Word followed by VS in questions, which is decidedly Germanic (a remnant of V2 word order). V2 occurs in some set expressions as well.
The placement of adjectives with nouns is also decidedly Germanic.
The English verb system relies heavily on helping verbs. A number of those are modal auxiliaries. Many Germanic languages have similar modal verbs to English (although they are often full-fledged verbs instead of being helping verbs).
The French verb system is very different, comparatively. The number of tenses, moods, and aspects and how they work is very different. Verbs serving modal functions are full-fledged, fewer in number, and often correspond poorly with Germanic ones.
English nouns have two grammatical cases (common and possessive). Proto-Germanic had many more. French nouns have no grammatical case at all.
Some English personal pronouns have two other cases (subjective and objective). Some French ones have remnants of the old Latin case system, not corresponding very well to Germanic ones (simple subjects, direct objects, indirect objects, disjunctive -objects of prepositions, compound subjects, emphasis, whenever the verb is not stated).
Some people might be tempted to claim that English using -s or -es to form most plurals is due to French influences (which adds a silent -s for most plurals), but that's far from certain. -s also occurs in other Germanic languages, like German (among many other patterns for forming plurals). It's possible that English chose to rely more on -s/-es due to French, but it may just be coincidental. Regardless, -s itself is Germanic as well.
studied linguistics and the history of languages, including English; taught French; native English speaker.