If you are seeing this effect after the jars have been in storage for a long period, do not eat the contents! This is a sign of botulism due to improper canning; the bacteria often (but not always) produce gas as they grow spores.
If this is happening immediately after the canning process, it is probably because you are not creating a proper vacuum seal. There are three accepted methods for doing this: Thermal exhaust, mechanical sealing (i.e. using a chamber vac) and steam displacement AKA steam injection. See the link for more information; the second two require specialized equipment, so thermal exhaust is what's normally done in a home setting.
In the thermal exhaust method, you get the contents very hot (71-82° C), which causes them to expand and release gases (air and carbon dioxide). After sealing and cooling, the subsequent contraction will create a vacuum seal. This is the most common method of home canning.
If you dump in hot brine and immediately seal the container, you are doing the opposite of this. Since the heat takes a while to distribute, you are causing this initial expansion when the jar is already closed, and this will force more gases up into the headspace and probably pop or warp the lid; if you're unlucky, it might even break the container.
Is this method safe? No. How unsafe it is depends on the acidity. Low acid foods must be processed in a pressure canner. That includes the majority of vegetables and especially garlic, peppers, etc. The main reason is botulism; the anaerobic environment is perfect for the C.botulinum bacteria, but spores cannot grow in high-acid environments with pH < 4.6. If the food is high-acid or has been acidified (i.e. pickled) then you don't need to worry, but with low-acid food you must kill all the bacteria and spores, and botulism spores are extremely heat-resistant.
In fact, they are so heat resistant that you cannot even kill them reliably in boiling water. That is why you need a pressure canner, to get the temperature all the way up to 121° C / 250° F, and you need to hold it there for at least 3 minutes. That is the only way to safely can low-acid food. Simply boiling it isn't good enough, and pouring in some boiling brine definitely isn't enough.
For high-acid foods, such as jams, a hot water bath is OK. This still involves boiling the entire jar after it's been closed, but you don't need a pressure canner. Pouring boiling liquid into a jar of cold or warm food will still not get the food up to a sufficient temperature, and even if it did, you would still need to sterilize the jar itself; that's done by boiling the whole apparatus. In this case you're not worried about botulism (since it can't grow in those conditions), but you still have to kill the other kinds of household bacteria, which don't have heat-resistant spores.
Since canning is intended to preserve the food, i.e. for long-term storage, you have to be a whole lot more careful about bacteria. You can't leave any opportunity for it to grow. Lots of people can the way your wife does and don't get sick, but it is a risky proposition, and I would strongly recommended you use safer methods, especially if guests or children are involved.
In answer to your second part, no. If something is not acidic enough for water canning it seems it doesn't matter how long you process it, it isn't safe. This is the reasoning all the cookbooks I've read provide for adding a teaspoon of lemon juice to each pint of tomatoes.
I would therefore use this strategy:
- get things to the right pH for water or pressure canning as a separate first step. (Fruit will generally be ok, tomatoes need lemon juice, pickles will be ok, etc)
- check the processing time for pints and quarts of various things (jams, fruit, pickles etc) and look for the pattern: is it consistently 25% more time, or 5 extra minutes? Work out the pattern.
- repeat for hot vs cold pack: do you halve the time, or subtract 10 minutes, or what?
Now armed with a hot-pack pints recipe, you can adjust it for cold-pack quarts or vice versa. You should also, in poring through these recipes and charts, have come to understand various categories (fruit, jam, pickles, nonpickled vegetables etc) and be familiar with what times they need. Then faced with a chutney, conserve, or pickle that isn't on anybody's list, you should be able to choose what category it most likely belongs to, and safely pick a time for it.
Best Answer
It sounds to me like the seal on the lids may be very slightly compromised. For example:
Edit
A thought occurs: it's equally as likely, if not more so, that the opposite of what I described above is happening: i.e., the stock is cooling down and contracting, which creates a negative pressure on the jar, sealing it. However, in a similar fashion to the above, a slight imperfection in the seal holds to a certain level of pressure but gives way past that threshold, allowing the pressure to equalize (which causes the top to pop up).
If they settled in the lid-down position after they completely cooled, they are probably okay (note the probably). What causes a lid to 'pop' after it cools is usually due to bacteria being sealed inside the can and/or compromising that can afterwards. Bacteria starts to eat the food, produces gas which expands and POP goes the lid, and your delicious canned food is no longer safe to eat. So if the lids stay down until you open the jar and the stock still smells okay, you're probably safe.