With respect to my colleague, I disagree with a lot of his answer. The core analysis I think is correct, not least because it comes straight from the GPL FAQ: if the plugin is tightly coupled to the main body of code, the whole (application and plugin) is a single work, which is a derivative of (amongst other things) the plugin; if not, then not. Note that this doesn't hinge on distribution arrangements: if you wrote your code to be tightly-coupled to someone else's pre-existing GPL code, then your code - from the moment of its creation - is a derivative of this GPL code, and when you distribute it, however you do that, GPL obligations will come into force.
Note also that because the plugin is free software, it will be very obvious to the plugin's authors whether or not the coupling is tight or loose. If it's tight, for example because complex memory structure is exchanged, then it would be very easy to establish that you had reference to the plugin's source during creation of your app, because there's no other reasonable way you could have discovered the API.
That said, I think the answers to your questions are as follows:
Do we violate GPL if we ship MIT licensed application pre-installed with GPLv3 plugin from the marketplace and do not release source for MIT licensed application ?
If the works are a single work as per the analysis above not only must you provide the source for the main application, but you must also convey the whole thing under the GPL. You can still release your application alone, without plugin, under MIT, but you must provide the source; and when the works are conveyed together, they must be free.
If the works are two works, it makes no difference whether they ship together or separately. If you ship the GPL plugin, you will still have source obligations in respect of the plugin, but that's all.
Is it acceptable to not assemble plugin (GPLv3) and application (MIT) prior to distribution and ask the customer to do install plugin (GPLv3) after the distribution ?
Asking the customer to assemble the pieces is fine, the licences are fine, not providing source for your app (if tightly-coupled) is not; see above. Copyright obligations are not a function of shipping arrangements.
Would the situation be any different if we had re-licensed our MIT app to a proprietary licence ?
No. It is the GPL that is placing obligations on you here, not the MIT licence.
An underlying confusion here is that you're writing a work and distributing under an MIT licence, without source. This seems to be causing problems, as some people are very used to being able to treat non-copylefted free code as being even more free than copylefted code, and thus are confused by your intent not to distribute source from the get-go. I maintain this is a cognitive error on their parts. It's completely lawful to distribute an original program under these terms, though it is unusual.