Exception to the ruling of striking the face

Not necessarily a comprehensive list, but some possible exceptions include:

• In war against disbelievers. This has been claimed as an exception by a faction of the scholars in their commentaries on the hadith, like here and here. This is taken from the fact that the full wording of the hadith on prohibition of hitting the face is specific to a brother , i.e. a Muslim.

  Among other evidence for that is also the apparent meaning of the Quranic verse:

  فاضربوا فوق الأعناق

  So strike them over the necks

  ^{— Quran 8:12}

  وقيل : للمؤمنين ... وقال محمد بن يزيد : ... المعنى أنهم أبيح لهم ضرب الوجوه وما قرب منها

  And it has been said that it is a command to the believers. ... And Muhammad bin Yazid said: ... The meaning is that it is permissible to strike at the faces and what is near to it.

  ^{— Tafsir al-Qurtubi}

• In Qisas. According to some jurists (this is a minority view) retaliation takes place for a slap to the face even when it does not cause a visible injury.

  وأقاد أبو بكر وابن الزبير وعلي وسويد بن مقرن من لطمة‏

  Abu Bakr, Ibn Az-Zubair, 'Ali and Suwaid bin Muqarrin gave the judgement of Al-Qisas (equality in punishment) in cases of slapping

  ^{— Bukhari}

  While there is agreement that there is retaliation for a proper injury to the face.

  والعين بالعين والأنف بالأنف والأذن بالأذن والسن بالسن والجروح قصاص

  An eye for an eye, a nose for a nose, an ear for an ear, a tooth for a tooth, and for wounds is legal retribution.

  ^{— Quran 5:45}

• In dealing with an intruder. According to some madhabs it is permissible to strike at the face of an intruder who is peeping into one's house without permission.

  لو أعلم أنك تنظر لطعنت به في عينك

  Had I known that you were looking I would have pierced your eye

  ^{— Bukhari}

  جاء ملك الموت إلى موسى عليه السلام . فقال له : أجب ربك قال فلطم موسى عليه السلام عين ملك الموت ففقأها

  The angel of death came (unannounced, in human form) to Moses عليه السلام and said: “Answer the call of your Lord.” Moses عليه السلام (did not recognize him and) slapped the eye of the angel of death and put it out.

  ^{— Muslim}

The hadith you quoted is itself about fighting, including where the intention is to kill, i.e. warfare, qitaal:

وإذا قاتل أحدكم فليجتنب الوجه

"fa idha qaatil a7adukum fa'lyajtanib al wajh"

"when one of you fights, avoid the face"

Further explaining this, from islaqa.info:

Al-Haafiz said: “This prohibition also includes all those who are struck for the purpose of hadd or ta’zeer punishments or discipline. According to the hadeeth narrated by Abu Bakrah and others, which was recorded by Abu Dawood and others, about the woman who had committed adultery, the Prophet (peace and blessings of Allaah be upon him) commanded that she should be stoned to death, and said, ‘Stone her, but avoid the face.’ (Narrated by Abu Dawood, 4/152). If that is the command in the case of one who is being punished and is going to die anyway, then the rule is even more applicable in cases of lesser severity.” See al-Fath, 5/216

So, striking the face is not allowed; whether in warfare or applying the hudud, which are obligatory; never mind striking the face in sports practice.

Turning to warfare; where you say:

Of course, it is safe to assume that during self defense and war it's about life and death and one surely can't help but strike the face. What can we say of such a situation? If such a situation is approved as an exception to the ruling, then would it be forbidden to practise one's combat skills (for either self-defence or war) which includes punching the face?

The rule about avoiding the face still applies. Bows and arrows, lances and the catapult (at Ta'if) were used at the time of the prophet saw without restricting them because they may have hit someone in the face; but this is not an exception, because what is aimed at isn't the face; if a rock thrown by a catapult hit someone in the face that doesn't contradict not aiming at the face, because aiming at the face or not is the choice of the one who aims; while where the blow lands is the fate of the one aimed at. So as long as the arrow or lance or catapult isn't aimed at the face then it's allowed and there is no exception to the rule, so your assumption isn't correct and the argument:

such a situation is approved as an exception to the ruling

isn't correct I'm afraid.

About the second part of your question:

Could we argue that striking the face is halal, in combat sports training, as long as we make sure we don't harm our partner (i.e. wearing protective headgear and trading light punches)?

As mentioned in the comment from @Medi1Saif, there are other posts here on this, and it seems boxing as a sport is unanimously and emphatically not allowed: see this post and this post.

In the last article, it is mentioned that the practice may be modified to not include hitting the face, which is actually what some clubs do; some street fighting and martial arts clubs run by muslims don't let people aim at the face or use shadow boxing or maintaining a distance; for proper boxing as a sport, even training for it, if sparring with a partner you are not allowed to hit the face regardless of whether a face protector is used or not. I have known brothers and family members who gave up boxing because of this, and they switched to other martial arts; equally there are clubs where they don't practice any actual hitting of the face, people do it for fitness (and the fact that body blows in street fighting are just as effective if not more so). But if you want to say, is it ever under any circumstances allowed to strike the face, then no, I never came accross anyone who ever said it was halal.