I doubt there are degrees of 'obsolete'. However, interpreting your question as I assume you intended it, there is a process of obsolescence (growing or becoming obsolete), which may be more or less advanced. In the case of 'muchly', that process is ... greatly advanced.
The OED does not mark the principal sense of 'muchly' (adv.) as obsolete:
1.
a. Much, exceedingly, greatly.
Beneath that sense, the editors have added in small type this annotation:
In later use usually with conscious humour, or for some other deliberate effect.
They do say "usually", and are careful to observe "or for some other deliberate effect". The effect need not be humorous.
For that sense, historical quotes are given dated to 1988. The lack of context does not allow me to judge with any degree of certainty whether 'muchly' is used humorously or not, or what other "deliberate effect" might have been desired. In regards to this last, the "deliberate effect" desired might well be to convey a (somewhat archaic, but not obsolete) conventional sense of formality and politeness.
Sense 1b is also not marked as obsolete:
b. humorous. ta (also thanks) muchly and variants: thank you very much.
And, while marked as humorous, the humor is light. For this sense, quotes are given dated to 1996.
From this evidence I infer that the obsolescence of 'muchly' is advanced, but not yet terminal.
For sense 2, however,
†2. Chiefly, largely. Obs. rare
the process is complete. 'Muchly' in this sense is obsolete. The only quote given is dated 1882.
[Definition from "muchly, adv.". OED Online. September 2015. Oxford University Press. http://www.oed.com/view/Entry/123141?redirectedFrom=muchly (accessed September 14, 2015).]
So, lastly, the answers I have for your non-titular four questions are these:
- "Are our usages acceptable within 'Standard English'?"
Yes, I think the uses you exampled are quite acceptable.
- "How common is formal use of the word?"
I'm not sure what is meant by "formal" in this context, but anyway, not very common at all.
- "What about informal use?"
Again, not very common, but, largely speaking, I suspect "informal" use is more common than "formal".
- "When did this word stop being normal regular everyday English?"
That's a tough question for me to answer on the basis of the evidence readily available to me (I'm lazy), but from the historical evidence I have looked at (comparing the historical adverbial use of 'much' in the sense given for 1a of 'muchly', that is, 'greatly'), I'm guessing the process of obsolescence began in the late 19th century and picked up pace through the middle of the 20th, then leveled off in its current state.