I know English was heavily influenced by French. But were there any other instances during which a Germanic language obtained Romance qualities or a Romance language with Germanic qualities?
Mutual influence between Germanic and Romance languages was huge. Some words crossed the language boundaries several times, e.g. the English word guard was originally Germanic (*ward), but English borrowed it from French.
There was also mutual grammatical influence; the passé composé of French and its Italian counterpart were modelled after the periphrastic perfect tense in German. On the other hand, Germanic languages completed their tense system after the Romance (Latin) model.
Last year I watched on TV here (England) a Flemish drama series sub-titled in English - I was amazed how many French words were used in conversational Flemish - the characters even said 'oui' for 'yes'.
Bavarian is a Germanic language, but it was heavily influenced by Romance languages. It's population started out as a mix of Romanized Celts, Germanics and partially Romans. This makes it unclear to which extend this early form of Bavarian was Germanic, but I think it's likely that it had many Latin words. Later on it was influenced especially by French (notably in the Napoleon era).
We have many words like "semme" (from Latin, originally meaning "wheat flour", now meaning "bun"), "servus" (from Latin for "slave" or perhaps short for something like "at your service"), "parasol" (from French, meaning, well, "parasol"), "laggl" (insult, coming from the French general "Ezéchiel de Mélac") or "maroni" (from Italian, describing edible chestnut). We also have some words like "mare" (compare to Italian "mare", German "meer" for "sea"), which I'm unsure about if they come from Romance languages or were simply a re-evolution of that word.
Though, sadly, today most of these words get replaced by German words.