OP: May a monastic regulate their alms diet in order to focus on healthy foods and avoid unhealthy ones?

The Theravada Bhikkhu Patimokkha states:

39. There are these finer staple foods: ghee, fresh butter, oil, honey, sugar/molasses, fish, meat, milk, and curds. Should any bhikkhu who is not ill, having requested finer staple foods such as these for his own sake, then consume them, it is to be confessed.

Based on this, I assume that it may be possible for a monk to request for healthier meals based on his health condition, but he may not get it, if he is dependent on almsfood.

In this article, we read about the 14th Dalai Lama:

During his early years in Tibet, the Dalai Lama would have eaten meat as part of his diet because of the country's reliance on consumable livestock in lieu of sufficient vegetable crops. When he fled to India in 1959, he adopted a lacto-ovo vegetarian diet in accordance with the beliefs of many of the country's own Buddhists. Unfortunately, after 20 months, he contracted hepatitis and developed gall bladder disease, and his doctors insisted he resume eating meat for health reasons.

Of course, the Dalai Lama is a popular religious leader who could easily get any kind of food he desires.

The Buddha also gave advice in the Donapaka Sutta (also see this answer) to eat less or eat moderately:

When a person is constantly mindful,
And knows when enough food has been taken,
All their afflictions become more slender
— They age more gradually, protecting their lives.

OP: Would it be appropriate for a monastic to physically exercise to keep healthy?

Please watch Ven. Yuttadhammo's "Ask A Monk: Physical Exercises" Youtube talk. He basically says that monks eat less than usual, do walking meditation and other physical tasks etc. and this is sufficient for monks.

Also he says monks should not be averse to suffering from health problems as it is a learning opportunity to understand reality.

The suttas also speak of monks exercising, for e.g. from the Meghiya Sutta:

Then in the early morning, Ven. Meghiya adjusted his under robe and — carrying his bowl & robes — went into Jantu Village for alms. Having gone for alms in Jantu Village, after the meal, returning from his alms round, he went to the bank of the Kimikālā River. As he was walking up & down along the bank of the river to exercise his legs, he saw a pleasing, charming mango grove

OP: Do monastics have access to healthcare for dental & physical checkups as well as brush their teeth, cut their nails, etc.?

I suppose they do, but I don't have info on this.

OP: May a monastic regulate their alms diet in order to focus on healthy foods and avoid unhealthy ones?

The Theravada Bhikkhu Patimokkha states:

39. There are these finer staple foods: ghee, fresh butter, oil, honey, sugar/molasses, fish, meat, milk, and curds. Should any bhikkhu who is not ill, having requested finer staple foods such as these for his own sake, then consume them, it is to be confessed.

Based on this, I assume that it may be possible for a monk to request for healthier meals based on his health condition, but he may not get it, if he is dependent on almsfood.

In this article, we read about the 14th Dalai Lama:

During his early years in Tibet, the Dalai Lama would have eaten meat as part of his diet because of the country's reliance on consumable livestock in lieu of sufficient vegetable crops. When he fled to India in 1959, he adopted a lacto-ovo vegetarian diet in accordance with the beliefs of many of the country's own Buddhists. Unfortunately, after 20 months, he contracted hepatitis and developed gall bladder disease, and his doctors insisted he resume eating meat for health reasons.

Of course, the Dalai Lama is a popular religious leader who could easily get any kind of food he desires.

The Buddha also gave advice in the Donapaka Sutta (also see this answer) to eat less or eat moderately:

When a person is constantly mindful,
And knows when enough food has been taken,
All their afflictions become more slender
— They age more gradually, protecting their lives.

OP: Would it be appropriate for a monastic to physically exercise to keep healthy?

Please watch Ven. Yuttadhammo's "Ask A Monk: Physical Exercises" Youtube talk. He basically says that monks eat less than normal laypersons, perform walking meditation and other physical tasks, and this is sufficient for monks.

Also he says that monks should not be averse to suffering from health problems as it is a learning opportunity to understand reality. But this is his opinion only.

The suttas also speak of monks exercising, for e.g. from the Meghiya Sutta:

Then in the early morning, Ven. Meghiya adjusted his under robe and — carrying his bowl & robes — went into Jantu Village for alms. Having gone for alms in Jantu Village, after the meal, returning from his alms round, he went to the bank of the Kimikālā River. As he was walking up & down along the bank of the river to exercise his legs, he saw a pleasing, charming mango grove

OP: Do monastics have access to healthcare for dental & physical checkups as well as brush their teeth, cut their nails, etc.?

I suppose they do, but I don't have info on this.