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This will usually happen with a lack of light. Are they also very light in color? This will further indicate that it is lacking light.

It is naturally a vine, so it will lay over without support, but if it is too weak then fixing the lighting will take care of that.

This article will give you information on the care of these: https://www.gardeningknowhow.com/ornamental/vines/mandevilla/

It really depends on if this is liquid or slow release fertilizer. Liquids will have their own instructions, but I prefer to feed at half strength every two weeks. If this is granular, or slow release, then once in spring will be fine each year.

Too much or too little water can cause mandevilla buds to fall off. The soil should be kept consistently moist, but not waterlogged (if the soil gets too soggy there may be a drainage problem with the soil mix). Pest problems can also cause this- check inside and around the flowers for tiny insects such as thrips or aphids.

Can you include a photo? This will help me to see what is going on.

This article will give you more information on their care: https://www.gardeningknowhow.com/ornamental/vines/mandevilla/mandevilla-plant.htm

Yes, you will want to add a support to your container.

https://www.gardeningknowhow.com/ornamental/vines/mandevilla/mandevilla-plant.htm