You already quoted 5:1 but if you go further you can read:
... Lawful for you are the animals of grazing livestock except for that which is recited to you [in this Qur'an] - ...
So we are told that it would be revealed in the Quran what kind of flesh is lawful for us!
In 5:3 you can read the kind of flesh which is prohibited:
Prohibited to you are dead animals, blood, the flesh of swine, and that which has been dedicated to other than Allah , and [those animals] killed by strangling or by a violent blow or by a head-long fall or by the goring of horns, and those from which a wild animal has eaten, except what you [are able to] slaughter [before its death], and those which are sacrificed on stone altars, and [prohibited is] that you seek decision through divining arrows.
and stunning is not one of those, which are prohibited, maybe through some explanation we can add it, but for now we look for a work around not for how to create a fatwa for prohibiting flesh!
In 5:4-5 you can read what is allowed for us and more exactly in 5:5 you'll find:
This day [all] good foods have been made lawful, and the food of those who were given the Scripture is lawful for you and your food is lawful for them.
Which says we are allowed to eat the food of people of the book. note that surat al-Maidah is one of the last revealed surahs and this verse is quoted among the last verses revealed.
I think this was the opinion of ibn 'Arafah (al-maliki) long ago and it was lately adapted by al-Azhar scholars and spread in Europe by them. I'm not sure whether al-Qaradawi has written the same in some of his books or given a similar fatwa. The addition made by al-Azhar scholars AFAIK is: If to say bismiallah Allahu akbar before eating the meat even if it was not slaughtered the correct way.
Some more fiqh details based on this fatwa:
If stunning happens before the slaughtering then there are three cases:
- if the animal was stunned and immolated before it dies, then it is halal.
- if it died through the sunning, it is haram as it is a dead animal (5:3).
- if one doesn't know, then one should take the overbalanced view, so if for example it was known that most animals die before they have been immolated than it would be haram if not halal.
This fatwa affirms that meat from slaughtering of people of the book is halal and also says that stunning from an Islamic perspective is haram (note that the word الصعق for stunning here seems to be understood as electrocution -which is only one of the methods described in the ruling-. The whole regulation seems to go ahead with Islamic teachings in avoiding harming the animal, and the purpose of stunning in this context is a kind of anesthesia, which shouldn't be as harmful as assumed in the fatwa), but if one can only slaughter this way it would be valid to do so.
This is so far an answer for the fiqh source and views.
Short answer for your other questions
Note that even in case of a Muslim slaughterer the animal could die from fear, or have a bad death if the person who is performing it was not experienced enough.
Also note that these rules of the EU apply only in case of a regular slaughtering that means for slaughtering for further sale! In this case usually a veterinary is present in would control everything so the chance of an animal dying for the quoted reasons isn't high!
I know that many Muslims buy animals directly from farmers and perform the slaughtering on site (without consulting a veterinary, this would be halal slaughtering, but is it safe?)!
Note also that many supermarkets in Europe sell meat which has a sign saying it is halal. So one can buy the meat there also many Muslim supermarkets have their own slaughterhouse or for example in Germany used to buy or import some kinds of meat from countries where it is/was allowed to slaughter the Muslim way (Austria was one of them a few years ago).
So there are options to buy.
Of course if one doesn't trust all of this we can eat and buy fish or become a vegetarian ;)
Issue of Odhiya الأضحية or 'Aqiqah العقيقة
This again is a workaround that I usually practice: the 'aqiqah for my children doesn't take place in our presence but at the house of either my parents or my in-laws, so for example my father in law buys a cheap and performs the 'Aqiah in our name (he could be refunded but usually doesn't accept it, but we also leave some money with him). One can buy a cheap for odhiya via a Muslim organization who spends the whole odhiya in a poor Muslim country or to people who would be in need.
But as said above I also ordered at a Muslim grocery here a cheap for 'Id and asked them to slaughter it in my name and got it from them on the day of 'Id or a day later. So in any case if you don't perform the slaughtering yourself you would be dependent on the person who performs it and you either have to trust them or don't in that case there's no way out expect buying an animal and slaughtering it yourself or as stated let others do it for you in a Muslim country as agents.
Finally I would like to point at a chapter in the Regulation
Derogation from stunning in case of religious slaughter taking place in slaughterhouses was granted by Directive 93/119/EC. Since Community provisions applicable to religious slaughter have been transposed differently depending on national contexts and considering that national rules take into account dimensions that go beyond the purpose of this Regulation, it is important that derogation from stunning animals prior to slaughter should be maintained, leaving, however, a certain level of subsidiarity to each Member State. As a consequence, this Regulation respects the freedom of religion and the right to manifest religion or belief in worship, teaching, practice and observance, as enshrined in Article 10 of the Charter of Fundamental Rights of the European Union.
I hope i covered your questions if i missed any feel free to remind me I'll edit my answer!