Gracias a todos por sus respuestas. Dejé que la curiosidad me venciera y, de hecho, la probé por mí misma (lo que estaba tratando de evitar en primer lugar). Espero que otros encuentren útiles estos resultados.
Como puede ver, con la excepción de IE, Firefox y Chrome advierten sobre el uso débil del algoritmo de hash, pero ignoran completamente el problema de las claves de 1024 bits.
Tested July 3, 2017
Case 1: 1024bit key, MD5, no subjectAltName
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Firefox = Warning that cert is hashed with disabled algorithm (MD5)
Chrome = Warning that cert is missing subjectAltName
IE = no warning
Case 2: 1024bit key, MD5, subjectAltName same as subject CN
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Firefox = Warning that cert is hashed with disabled algorithm (MD5)
Chrome = Warning that cert is hashed with weak algorithm (MD5)
IE = no warning
Case 3: 1024bit key, SHA256, no subjectAltName
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Firefox = no warning
Chrome = Warning about certificate missing subjectAltName
IE = no warning
Case 4: 1024bit key, SHA256, subjectAltName same as subject CN
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Firefox = no warning
Chrome = no warning
IE = no warning
[1] Browser versions tested:
* Firefox 54.0.1
* Chrome 59.0.3071.115
* IE 11.0.9600.18698
[2] All tests performed on Windows 8.1 32-bit OS
[3] All server certificates signed by a self-signed Root CA,
with no Intermediate CA. Root CA certificate was installed
into the Windows certificate store using mmc.exe, and
Firefox using the Options GUI.