Under doesn't make sense in either sentence. The way your first example reads is that "QA system" is some kind of image, and that there is supposed to be a background image appearing below it, but it doesn't show up. But since a QA system is not an image, that doesn't make any sense. In doesn't really work either. It doesn't make sense to be missing an image in a system. You can be missing an image on the QA screen or on the QA page. I can't 100% determine what you're trying to say here, even after reading comments on the question. But maybe you mean something like this:
The background image is missing on the QA page.
This means that in your application there is a page dedicated to QA, and this page's background image has failed to load.
Your second question is pretty different; I know both are re: "in vs under", but since neither in nor under make sense in the first case, it's not really the same. Regardless, the correct word to use with weather is in:
I wouldn't jog in this weather.
We go out in the rain and out in bad weather. Once you're outside, you're in whatever type of inclement weather there is. Under doesn't make sense. You aren't under rain, for example; the rain is falling all around you. You're in the midst of it.
This is a good question.
IMO, you don't need anything in that sentence though. And, I'm a non-native speaker. Though the question is not asked to us, I'd like to express what I think about the use of 'when' and 'while'. What is preferred by native speakers, they'd only answer.
Smartphones allow Internet access on the road - seems okay to me!
But then, if it's about putting while or when...
Both 'when' and 'while' indicate time-related actions. Both the words are quite often used in sentences with past progressive and simple past tenses. In such cases, while generally refers to a comparatively long action, on the other hand, when could be both a comparatively long or short action. Also, while has a clause with a simultaneous activity or ongoing activity.
In that sense, both seems okay to me. Smartphones allow Internet access while/when on the road.
Nevertheless, when you are going from one place to another place, your smartphone will get you the Internet access while on the road. And, when your friend asks whether smartphones allow access, you may simply nod and say that smartphones allow Internet access when on the road.
Best Answer
Your question is slightly misleading as you refer to input in the title, but then ignore it is the body of the question.
Regardless, you can not use on with the verb to input, but use can use both in and on with the verb to put.
Therefore, you can
or
Either is correct, and commonly used, although I would argue that on is more likely to be used in an informal sense or by non-technical people.
However, you can only