Can a word like violence, which is an uncountable noun, be made countable? For example, there are different types of violence such as physical violence, emotional violence, etc. In this instance, would violence be countable?
The brief answer is "yes", but this has to be qualified by a historical viewpoint.
There are many references to "these violences" in the period before the 20th Century. The use of the phrase declined thereafter and occurs little now.
Here is ngram's graph:
One of many secular examples:
The French persisting in their claims to the country on the Ohio, as part of Canada, strengthened themselves by erecting new forts in its vicinity, and at length began to seize and plunder every British trader found on any part of that river. Repeated complaints of these violences being made to the Governor of Virginia, it was at length determined to send a suitable person to the French commandant near the Ohio …
One of many religious examples:
These Violences were done all Scotland over , in such Places where the Presbyterians were absolute Masters , but with best Success upon the South Side of the River of Forth.
and a contemporary example:
These violences care not for national boundaries and also tend to most affect those who are already the most deprived.
This all suggests that, although a little archaic, you might be justified in listing the sorts of violence (physical, emotional, financial, institutional, wartime and so forth) and then writing similar prose on the lines of "These violences constitute an affront to humanity".
Can a word like violence, which is an uncountable noun, be made countable?
There are very few words that are only uncountable and probably none that are only countable. Uncountability / countability is usually, (but not always) an attribute of nouns within context. And that attribute may be present or not.