Several references that the Grammar Exchange consulted did not note a difference between "weep about" and "weep at." In your sentences, it would be possible to interchange "about" and "at," with the essential difference being one of timing.
In the first sentence, if the ministers "wept at" the death of the military colleagues, it would mean immediately upon hearing the news. In the way you have the sentence, the weeping could occur at a later time, for example in recalling the death.
In the second sentence, if the person were to "weep about" a dog's death, the weeping could take place at some time after the dog had died, when remembering the dog. "Weep at a dog's death" would be an immediate reaction to learning that the dog had died.
"Weep at" appears in the Collins online concordancer*:
1) ...her so that I did stop, for she would weep at the mention of you after that. `
2) ...the impact of her Versace frocks, and weep at the loneliness of the poor,
3) ...Few people in Britain will weep at the defeat of Paul Keating, the
Only sentence 1) shows that the weeping takes place immediately at the mention. Before "mention," then, it appears that only "weep at" is appropriate.
In sentence 2) one could well, also, "weep about" or "weep for" the loneliness of the poor.
As well, in sentence 3) one could weep "about" the defeat of Paul Keating. However, "weep for" does not seem to fit. One might, however, "weep for" Paul Keating himself. I'm not quite sure why.
"Weep" appears with several prepositions, notably "for." Here are examples from a definition from the American Heritage Dictionary of the English Language**
"¢ To mourn or grieve: wept for the dead.
Under "cry," in the same dictionary, there is this example for "weep":
"¢ "I weep for what I'm like when I'm alone" (Theodore Roethke).
The Collins online has only this example for "weep about":
"¢ they wept easily: there was plenty to weep about. The regular tearful goodbyes
In this sentence, "at" can not be substituted, but "for" can.
The BBI*** dictionary has these examples:
"¢ to weep about; over (to weep over one's misfortune)
"¢ to weep for...
"¢ I wept at seeing them so poor
We can also "weep about, "weep for," and "weep over" one's misfortune, and "about," "for," or "at" seeing them so poor.
**The American Heritage Dictionary of the English Language. Fourth Edition. Houghton Mifflin. 2003
***The BBI Dictionary of English Word Combinations. Benson, Benson, and Ilson. John Benjamin Publishing Company. 1984