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Early morning or early evening would be the best time.

You can plant the plumeria in the ground anytime now, as long as you are able to keep up with watering as it gets established.

The best fertilizer for plumeria trees is one that is high in phosphorous, such as 10-30-10. A granular formula is preferable to liquid.
Fertilize the plumeria every three months beginning in April and ending in August if you live in an area with frosts. Otherwise, continue to fertilize year-round.

https://www.gardeningknowhow.com/ornamental/flowers/plumeria/how-to-fertilize-plumeria.htm

Plumeria will bloom whether or not pollinated. You only need the flowers to be pollinated if you want seeds, but most cultivars won't produce seeds anyway. Plumeria contains both male and female parts on the plant, so yes, it can self pollinate, but there is no guarantee. It more often is pollinated by insects. You also can pollinate the plants yourself.
Here is more information:
https://wimastergardener.org/article/plumeria/
https://www.gardeningknowhow.com/ornamental/flowers/plumeria/harvest-plumeria-seeds.htm

This is normal to some degree. This is a sign of rapid growth. Here is an article that will refresh you on their care requirements: https://www.gardeningknowhow.com/ornamental/flowers/plumeria/growing-plumeria.htm

We do not sell plants; we provide information to gardeners.
I would suggest contacting the seller you purchased from for a replacement.
This article has care information for you.

https://www.gardeningknowhow.com/ornamental/flowers/plumeria/growing-plumeria.htm