The Hobbits spoke Westron, the common tongue, but retained what little was left of their own language for things like personal names and days of the week:
And in those days* also they forgot whatever languages they had used before, and spoke ever after the Common Speech, the Westron as it was named, that was current through all the lands of the kings from Arnor to Gondor, and about all the coasts of the Sea from Belfalas to Lune. Yet they kept a few words of their own, as well as their own names of months and days, and a great store of personal names out of the past.
From the Prologue, part I., "Concerning Hobbits" (pg 4 in my copy).
(*: "those days" were when the hobbits had for the most part stopped their wanderings and had begun to settle in Bree, near what became the Shire.)
Bilbo being a proper name, it must have come from their old language. Knowing that the dwarven names come from the Elder Edda, and so forth, it is not unreasonable to assume there is a source for these proper names.
Letter 25 notes:
The hobbit-names [are] from Obvious Sources proper to their kind.
The question then, of course, is what exactly these "Obvious Sources" are. (The phrase "proper to their kind" is also intriguing as many have compared hobbits to the British servicemen in war who Tolkien so admired - see for example the work of Joseph Loconte.)
In Letter 72, Tolkien writes:
Hobbits of that class have very Saxon names as a rule
Which is quite enlightening. The dictionary notes of Saxon:
relating to the Anglo-Saxons, their language (Old English), or their period of dominance in England (5th–11th centuries).
Looking at the word origin of Bilbo, it seems to come from bilboa and bilbao, both of which are purely English in origin. Tolkien, as a professor of English, would probably been aware of these words, and might have been influenced by it, especially as Bilbo owns the sword Sting, of Elvish make (therefore a good blade).
However, there is another twist to this (thanks to the comments for pointing this out) - the "that class" here refers to hobbits of the class of Samwise and his Gaffer; Bilbo as a Baggins would not be of this class but a higher one, and thus this line crumbles a bit.
I will note that also in Letter 25 it says:
The language of the Hobbits was remarkably like English [...] Their family names remain for the most part as well known and justly respected in this island as they were in Hobbiton and Bywater.
This also seems to indicate that the proper names were English.
The orcish languages were not exactly consistent, as often the orcs of different regions would due to arguing come up with their own often very different dialects; the languages were often limited and angry, a true perversion of the elvish languages. It is therefore surprising to me that a possible source could come from English, as the orcish and black speeches may well have come from Hittite and Hurrian (see here), and it also seems odd that Tolkien would wish to associate the language of his native country with some of the most repugnant creatures in his works.
According to the Tolkien Gateway:
The meaning and origin of the name Azog is unknown. It is most likely a name in the Black Speech (e.g., the similarity between azog and nazg), though Mágol may also be a possibility. Andreas Moehn has noted an apparent but tentative link with Mannish, in the Adunaic word zagar meaning "sword", in which case Azog might refer to "warrior" or anything similar.
(Nazg means [finger]-ring or [magic]-ring.)
If it did come from Black Speech, as likely, that would indicate a very low probability that that is association with your 'azoch'/'azoth'. Adunaic, on the other hand, is more interesting. Adunaic was the speech of the Numenoreans, which derived to some extent from the Elvish languages (as the Numenoreans were those who were gifted by the Elves, so they were very aware of Elvish languages) though with some distinctions. For example, it has some of the characteristics of Khuzdul (the language of the dwarves), and also has noun classes. Westron then derived from Adunaic.
The language of the Rohirrim is an archaic relative of Westron similar to Old English, therefore by taking this route it is not completely unlikely that 'Azog' may have had its origins in 'azoch'/'azoth', but for in-universe reasons I am disinclined to believe this the case (as stated above).
There is yet a third possible route. The black speech (probably) was based upon, as said earlier, hittite and hurrian, and a not insignificant selection of orcish dialects were probably based upon this as the orcs were created (or rather perverted) by Morgoth, and Sauron was his lieutenant (see Silmarillion) but there were some orcish dialects that were based upon Westron, according to Robert Foster's Guide to Middle-Earth under 'Westron', though it should be noted that this book was published before the History of Middle Earth series and its associated information came out.
If the orcs Azog was among used such a Westron-based language, I would still, however, have some doubt in my mind as to whether this was an association Tolkien would make.
It then gets more confusing when one comes to the 'Orc' section in the same book. Here it says that the more Westron based orcish dialects were used for communication between tribes, whereas more black-speech based dialects which varied dramatically from tribe to tribe were used within tribes. If this is the case then the personal name of Azog probably still came from this black-speech based dialect; under 'Black Speech' it says that many Mordor orcs, indeed, had their names in a "debased" form of black speech; then again, Azog wasn't necessarily a Mordor-orc, though he could have been.
An important note
It should also be noted that Westron is represented as English in the book, but the whole book is represented as translated, so the names that we're referencing weren't the actual names, but the translated names, which I'll admit gets kind of confusing. For more information on this bit, see the font of all knowledge, Wikipedia or Appendix F of the Lord of the Rings.
To summarize the situation as best I can: since Westron was the most commonly spoken language in Middle Earth, Tolkien treated that language as English and then worked from there so that languages closely related to Westron, like the language of the Rohirrim, were treated accordingly. Thus the language of the Rohirrim comes out much like Old English. In other words, Tolkien as the translator represented this world as if it was ours and the legends in our tongues (sort of; his own versions of our tongues with a few others added in).
I would check the conlang stack exchange which allows questions on Tolkien's languages. I'd also take a look at this website (one of the most detailed and well-sourced descriptions I've found of Tolkien's various languages). Finally, check out the History of Middle Earth, especially, in the case of this question, volume 12, The Peoples of Middle Earth, which, to quote the above site,
The Peoples of Middle-earth gives many more "original" forms of the names Anglicized by Tolkien than the ones mentioned in the appendices to LotR.
A final small note: Tolkien didn't exactly like Shakespeare; he wrote the story of the Ents marching on Orthanc because he did not like the way the marching of the forest was portrayed in Macbeth; he also did not like Shakespeare's writing of Elves. Hopefully this helps.