Am I supposed to pick off the spent dianthus flowers? They are hard to remove.
You can shear off the top inch or so from the plant to deadhead it rather than try to take off individual flowers.
Due to last year's hot wet weather, my Dianthus now have an underbrush of dead foliage but the tops are still green. They are densly populated. I am thinking of shearing the undergrowth to save the plants as I have about 150 of them. Would this be advisable?
You can cut them back by about 1/2. This should encourage lower branching are refoliation.
Do they bloom through summer and how often do I need to feed them?
There's no need for pinching, though doing so will not hurt the plant. Yes, they will bloom throughout summer. You can apply a balanced fertilizer every 6-8 weeks or simply use a slow-release fertilizer at the time of planting, along with some compost. This should be sufficient and lessen the need for you to continually do it.
Also, deadheading is often a good idea once the blooms fade, as many varieties will self-seed. If you don't want them popping up in unexpected places, this will help. Otherwise, you can let them be and allow them to surprise you with additional plants.
Someone keeps digging it out when I look the next day.
Raccoons are notorious for digging up plants and then just leaving them, and they often come out at night. Squirrels are also known for digging up plants, both to bury and retrieve food. Most animals can smell when the ground has been disturbed and likely assume something tasty has been hidden there. If the plants are dug up at night, however, it is most likely raccoons. Squirrels aren't nocturnal and do their digging during the day and mostly stick to freshly planted bulbs rather than plants. This article will help with raccoons: https://www.gardeningknowhow.com/plant-problems/pests/animals/getting-rid-raccoons.htm
Why are the stems of my pinks drooping over?
Dianthus gratianopolitanus, Cheddar pinks, or in our area, "Pinks" are sun loving evergreen perennials. Given 6 or more hours full sun a day produces an erect stem. At times, the plants might get a little leggy, and need a trim of spent blooms, stems and top couple of inches of foliage.
Are Dianthus annuals or perennials in Central Mexico? Will they come back yearly?
Dianthus' habits are more related to the variety of plant than to where it's growing. This article has more information: https://www.gardeningknowhow.com/ornamental/flowers/dianthus/growing-dianthus-plants.htm
I'm growing baby doll dianthus from seed. The seed packet states keep in warm place. Gardening site states keep in fridge? Also, when should I transfer to garden? Many thanks.
Keeping seeds in the refrigerator is usually done to simulate winter dormancy, thus encouraging better germination. To be honest, I would always prefer gardeners' experience to directions on packaging; I've seen too many labels that are less than accurate. Perhaps when they say "warm place," they mean for actually germinating the seeds. Here's an article that may contain a few tips you can use: https://www.gardeningknowhow.com/ornamental/flowers/dianthus/growing-dianthus-plants.htm