Preheat the jars with hot tap water, and then just dump them in the boiling water (with tongs or other appropriate utensil, of course).
What you are doing is not sterilizing the mustard. You just pasteurize it.
If you just want to pasteurize the mustard: Yes, you can put the jars into the oven instead of water-bathing them. Jars, lids and the content are heated up to 100°C, so it's quite equivalent to water-bathing the jars - if not even better.
Friends of mine put the jars with the lid loose on onto the baking tray with some water (for the moisture in the oven to not to damage the gasket) and then "bake" this for 30-120 minutes (I can't remember how long exactly) at 150-200°C. Then the jars must cool down slowly, therefore the jar must stay for another 30 minutes in the oven which is turned off. I think the lids are closed when the jars are taken out of the oven (but I'm not sure).
After reading this text 1 I don't know whether to preheat the oven or not.
Note: In Germany people often can raw food! Since your mustard is not raw, you just need to sterilize the content of the jar and don't need to cook what's inside. Therefore the cooking/baking time should be decreased.
If you want to sterilize your mustard, you need weck jars with rubber seals and steel tension clips. When baking the jars with the contents and the lid on, secured with a rubber seal and the tension clips at more than 120°C, the contents are sterilized like in a pressure cooker. The rubber seal and the tension clip act as a valve. After baking, the contents are vacuum-sealed. See wikipedia article about home canning.
image by Alupus.
1
Das Wasser darf nur langsam erhitzt werden. [...] Je nachdem, wie weich das Einweckgut ist, dauert das Einkochen zwischen 10 und 120 Minuten. Danach dürfen die Gläser wiederum nur sehr langsam abkühlen. [...]
Statt auf dem Herd kann man auch im Backofen einkochen. Zum Einkochen im Backofen die Fettpfanne mit Wasser füllen, auf die unterste Schiene schieben und die Einweckgläser, wie oben beschrieben vorbereitet, hineinstellen. Obst bei 150 bis 160 Grad einkochen, Gemüse bei 190 bis 200 Grad. Die Gläser danach noch 30 Minuten im Ofen stehen lassen und anschließend zugedeckt abkühlen lassen.
translated freely:
[The first paragraph deals with canning with a water bath] The water must be heated slowly. [...] Depending on how soft the food the be canned is, the canning takes 10 to 120 minutes. After that the jars must cool down very slowly.[...]
Instead of canning on the stove you can also do this in the oven. Put the jars onto the baking tray and pour some water into the tray. Fruits are canned at 150°C to 160°C, vegetables at 190°C to 200°C. After that, the jars stay another 30 minutes in the oven [note: which is tuned off] and afterwards let these cool down covered up.
Best Answer
The right way to know if your recipe is safe is to use a recipe from a trusted source. What you're doing is pickling, so just search for pickled pepper recipes. For example, I quickly found this one:
http://nchfp.uga.edu/how/can_06/pickled_hot_peppers.html
The NCHFP is one of the big authorities on canning in the US, so you can certainly trust that. You can add spices, and if desired add more salt or vinegar, but you don't want to use less - it might become unsafe.
And follow their process - there are reasons for everything. For example, all the boiling and details about how to fill the jars help guarantee that you get a good seal, so your jars will stay safe long-term. What you're doing is really prone to bad safety issues. For example, things could get colonized by bacteria while you let them cool, then you'd seal it in the jar and let it go to town. Or your seals could fail. You may get lucky for a long time, but not everyone will every time.
You should be able to find plenty of similar things from other universities, if you need, or you can buy an authoritative cookbook/reference.