Sound on iPhone is reasonably complicated when you get down and start "taking it apart". This is because it is managed on a very much case-by-base basis.
For example, if you call up Siri, and adjust the volume, it will only be adjusting the volume for Siri, not something like the ringer volume.
So to divide things up and try to answer your question, I'll start with the basics.
If you turn on Vibration mode (orange position) using the switch on the side of the phone it will turn off almost all sounds. Unlock, keyboards and such sounds will no longer be on. Notifications and phone calls will vibrate but not be heard. You will have to turn the switch off (no orange) in order to hear normal sounds while in apps - (i.e. games). When alternating vibration mode versus normal mode, iPhone remembers your previous state, and will set the volume to what it was before vibration mode was active.
Exception: alarms set in the Clock.app are still heard. In fact, there is no way to turn off Clock.app's alarms from being heard unless you turn off the alarm(s) itself. As for other alarm clock style apps (for example, Sleep Cycle), it is up to the developer to make it behave like Clock.app. That is the case with Sleep Cycle, but it would be worth verifying with any other alarm style app.
If you turn down the volume in the Music.app (or double tap home button and swipe to the right twice), or while watching a video, or in Settings.app, that is the volume that is applied across the board, mostly. That volume reflects the volume for videos, music, and apps that in general make noise - as in, games. Note that iPhone remembers your settings when you have headphones plugged in vs. not.
When iPhone is unlocked and you are on the home screen and you adjust the volume, you are adjusting ringer volume. In other words if the vibration switch is off (no orange) then that would be the volume it would ring at.
Do-not-disturb suppresses basically all notifications (there are some configuration options there). Again, alarms (from Clock.app, and other properly set up apps like SleepCycle) will stay play.
The only thing you can't do is have it toggle sound settings when it is plugged into the wall vs. when it is not. I think you will find it becomes habit, however, to throw the vibration switch.
I've researched this topic myself, and everything I've found says that the HDMI port and/or Thunderbolt port on Macs is not meant as a display input. You can't use it for that purpose.
However, something that may work, if you're willing to spend money, is a capture card. I've used them for recording video game footage before. The one I use has a noticeable lag. It's split from the TV to the capture card, and what happens on the TV takes 1 or 2 seconds to display on the input from the capture card.
If you could find one without lag, or with very minimal lag, it'd be usable for gaming purposes. Of course, if you're trying to hook up something that doesn't need to be lag free, such as a movie player, or anything else that plays non-actively controlled media (actively-controlled media would be video games, and pretty much nothing else), you could do with a capture card that has some lag.
This is a capture card that advertises OS X compatibility: http://www.blackmagicdesign.com/products/intensity/
It has probably any input you could need. I'm not sure if it lags or not, as I have not used it. Also, you have to consider the fact that they may use proprietary software for recording. If they do, and there's no way to display a video stream without recording, it's probably not a good solution for you.
Some quick searching seems to indicate there is no noticeable lag with Blackmagic's Intensity capture cards. It also seems to suggest that their software can display without recording. I don't know if you can full screen it or not. Or if you can display a video stream in a program of your choice.
Edit: I just noticed on their site, they said it supports streaming. And doing so with "all leading streaming software" too. Which would suggest that it does provide a video stream, which you could utilize in a program of your choice, and also display in full screen.
Best Answer
AFAIK, this is definitely not possible. The closest thing to what you're describing would be AirPlay. However, AirPlay requires a newer Apple TV connected to a TV; it wouldn't work on an iPad or a Mac, let alone some cheap old Android tablet. Additionally, even if it were currently possible to mirror the display of your iPhone onto an iPad, that's all it would do is mirror the display of your iPhone; it wouldn't be able to respond to multi-touch.