Yes the Honda City would be very well able to glide over the speed bumps/humps on Indian roads. In fact, many cars currently on sale in India have the ground clearance of 165 mm like the Chevrolet Cruze. Also etc. Also cars like Skoda Octavia also come with a ground clearance of 155 mm which also face no issues with the speed bumps. So, that should not pose any issues with the Honda City.
Certainly, you can get your car modified as per your choice. Coming to the front grill, you can replace it with the different grill. For that, we suggest you to get in touch with the authorized service centre as they will be the best people to suggest you few compatible grills to choose from.
Click here to find the Honda service stations by selceting your city in the drop down list.