May a monastic regulate their alms diet in order to focus on healthy
foods and avoid unhealthy ones?
As a monk, you generally accept what you are given. However, after collecting the food, you can choose to eat certain foods from your bowl and discard other foods; although this may be awkward to do when you are given foods & eat in public (such as at a public ceremony).
If you live in a hardcore monastery, often all the alms food is mixed together in a large bowl and then ladled out; thus preventing choice. However, this is rare.
Unless you are 'rock-star' monk or nun in California, you are generally given food by Asian benefactors, such as white rice, curries and lots of sweets. It is quite noticeable many monks in Western monasteries supported by ex-pat Asians appear overweight (possibly due to lots of public offerings & public eating and being offered lots of cakes & sweets).
A monk came to live in my small local area (in Australia) once for a few months. You could see his craving for my food when he visited my house (where i provided healthy brown rice, lots of veggies, eggs, tofu, nuts, etc). Most people offered him rubbish to eat. But I expelled him after a few days because he taught Dependent Origination incorrectly (when unasked!!). Therefore, a monk with wrong views that slanders the Dhamma and the Buddha may also not receive healthy food but, instead, eat junk food from the hands of fools.
Would it be appropriate for a monastic to physically exercise for the
sake of retaining their health?
In a hardcore monastery, generally there is physical work to do, such as sweeping the forest paths, working on building projects, carrying water, etc. Plus there is alms round and walking meditation. Therefore, a monk exercises.
However, there is nothing stopping you doing exercises in private, such as doing push ups and sit ups in your hut. But visible exercise in public is generally disallowed.
Once I saw in Thailand a German monk kicked out of a monastery for insisting on doing Tai Chi in a public park in the monastery. The German monk cried it was unfair.
Do monastics have access to healthcare for dental & physical checkups
as well as brush their teeth, cut their nails, etc.?
Generally, yes. Monks are generally well looked after. When I lived in well-known monasteries in Thailand as a layman, the local hospitals always gave me free medical care when required, including once during a one-week stay in Bangkok.