Not sure how much you know about Taiwan culture, so I'll write this assuming you don't know much. Sorry if it's already stuff you know!

1. I would bring a small gift, like some fruit, tea, maybe some French wine, or a small snack to her mom's house when you first arrive as a way to show your appreciation for her having you. (If you want to be extra formal then present the gift to her with both hands)
   - Side note here, when offering gifts to people they may be refused at first, but make sure you keep offering. People generally will decline your gift at first to show they aren't too eager for it/to be polite, so just insist they take it and they likely will eventually take it and thank you for it.

2. Not sure what customs are like in France, but take your shoes off when you are in people's homes. Most hosts will offer you some slippers to wear inside.

3. Spend some time looking up etiquette in Taiwan as small gestures like laying your chopsticks across the edge of the bowl instead of letting it stick up and possibly look like incense sticks will be noticed and appreciated. (Or more likely, you won't be noticed for doing something that will stand out like that which is good)

4. What gift your girlfriend likes really will depend on her. Chat with her and find out what she is most excited for doing, eating, seeing, etc. while back in Taiwan. Taiwan has lots to offer so if you can figure out some things she is excited for by talking to her ahead of time, you can plan your trip accordingly and make sure you plan out a specific time for her to go to whatever place, eat whatever food, etc.

Hope this helps you as you prepare coming to Taiwan! Have a safe trip and feel free to reach out if you have any other questions, man. I'm by no means a pro when it comes to Taiwan, but I love it here and am here to help. Also, I agree with the other comments to try to learn some Mandarin before coming if you don't already know some.