The person's file shall be examined only based upon his violating or not the EU laws.
There are legal provisions that allow people who have broken certain laws (including people who have been charged and convicted of criminal offenses and even served jail time in EU up to a certain limit) to obtain residence permits for a number of different reasons.
For example, in Poland, being in Poland illegally is not solely a reason to reject a residence permit for a family member of a Polish citizen - this is what the law says, not something subject to what you or I think shall be basis of examination.
I do not believe the woman's health shall be taken into consideration in this matter
For example, in Poland, "other compelling reasons" shall
be taken into consideration when examining someone's application, and in this case, he has a particularly strong compelling reason which is a Polish citizen that is dependant on him to provide care due to health conditions. Several Ukrainians have residence permits in Poland exclusively
for that very reason.
Finally, EU law guarantees the right of EU citizens to live with their families (and by definition in each and every single EU country that includes the legally wedded spouse, and in some countries a civil partner as well).
Pretty much the only reasons an application can be rejected even on appeal are:
1. Person seen is a potential risk (usually terrorism links, organized crime links or subject to EU sanctions). This CANNOT be argued against.
2. Person has served a long jail term for a crime committed in EU (usually longer than 2 years). This can be argued against.
3. The marriage is proven to be a marriage of convenience. And if there is a child, this possibility is automatically eliminated - by law. This can also be argued against.
4. Person has been ordered to leave EU and failed to comply or legally appeal against such orders. This can also be argued against.