Let me deal with your questions one at a time:
Why should the Jewish people accept the Oral Law as authoritative if: 1)The oral law is not mentioned even once in the entire Tanach
There is actually at least one clear reference to the Oral Law in the Chumash (and, according to Rabbi S.R. Hirsch a startling passage of ORAL Torah within the Written Torah itself!):
"When the Lord your G-d will widen your borders as He promised you [i.e., bring you into Israel] and you will say 'I will eat meat' for you will desire to eat meat; according to all the desires of your soul, eat meat. When you will become distant from the place that the Lord your G-d will choose to place His name [i.e., Jerusalem and the Temple] and you will slaughter from your cattle and from your flock (that which G-d has given you) as I commanded you..." Deut. 12; 20, 21
Rashi: "'As I commanded you:' This teaches us that there is a (detailed) command concerning slaughter - how to slaughter - and these are the laws of slaughter that were taught to Moshe from Sinai (i.e., from G-d)."
So G-d, somewhere, transmitted detailed laws of ritual slaughter. I invite you to search the Written Torah from its beginning until its end and challenge you to find one single detail: there's no mention of what type of tool to use (knife, hammer, poison...) nor where on the animal to make a wound (throat, back of the neck, major arteries...) nor who should do it, nor when or where it should be done. Yet all of these details are necessary for the proper observance of this command.
We're left with two possibilities. Either the author of the Torah was a bit scatter-brained and left out some details (intending, no doubt, to get back to it later), in which case, the Torah's divinity - or even its coherence - becomes an impossibility. Or that the Author was in full control of His material and included these and countless other details in "footnotes." If there's one "footnote" there could easily be more...especially since there are so many passages in the Chumash that are so unclear by themselves (take the mitzvah of tefillin as an example: I defy you to make any sense of it from the text itself).
As to Rabbi Hirsch, see Exodus 15 from the beginning until verse 20 - it all falls under the superscription (verse 1) "and G-d spoke to Moses..." and contains many details of the Passover offering. Verse 21, however, reads "And Moses [*NOT* G-d] spoke to all the elders of Israel..." with more details of the divine command for which we have no explicit record!
2)When God told Moses to come up to Mt. Sinai He said: "come up to me into the mountain, and be there: and I will give thee tablets of stone, and a law, and commandments which I have written" no mention is made of an oral Law
Nor does there need to be any, as sufficient mention is made elsewhere (see above). However, you actually have a much bigger problem here: you (correctly) quote G-d, immediately after the revelation at Mt. Sinai, saying "which I have written" in the *past* tense. Yet, according to the account of the chumash, the Torah itself wasn't written until the very end of Moses' life (see Deut. 31; 24 - "and it was when Moses finished writing the words of this Torah in a book until their end"). So this passage can be referring to neither the written nor oral versions of the Torah as a whole.
Why not, then, understand it as does the Talmud (TB Brachos 5a): "...'the tablets' (refers to) the ten commandments, 'Torah' (refers to) the Written Torah, 'and the commandment' (refers to) the Mishna, 'that I wrote' (refers to) the Prophets and Writings."
3)The Tanach reports that the written Torah was both lost and completely forgotten for over 50 years and only rediscovered by the Temple Priests. How can the Oral Law have been remembered when even the written law was forgotten?
Now that's a question that also bothered me some years ago. Let's look at the language of the actual passage (II Kings 22; 8 et. seq.) "And Chilkiyahu the high priest said to Shafan the scribe: 'I have found a book of the Torah in the House of G-d'...and Shafan the scribe said to the king: 'Chilkiya the priest gave me a book' and Shafan read it before the king..."
Now there is no indication at all that there were no written Torahs in Israel over any period - only that the discovery was noteworthy and was immediately referred to the king. Why was it noteworthy? Most likely because Yoshiyahu the king's great-great grandfather (Achaz) and his grandson and great-grandson had burnt some scrolls in their attempt to limit Judaismâ€™s influence on their monarchies. Finding, therefore, an intact scroll in such a public building as the temple itself was a surprise. But there is absolutely no indication that much more damage had been done beyond whatever was locally available. Certainly not universal destruction and a subsequent ignorance. And to the contrary, an Oral tradition would stand up much better to book burnings than the books themselves.
The young king himself would of course have been surprised. Having been raised in a royal court devoted to the negation of Torah, he would never have come in direct contact with these words and he took them to heart. Other Talmudic traditions, by the way, maintain that the volume in question was the original scroll penned by Moshe himself (see II Chron. 34; 14) or, alternatively, happening to be rolled to a passage of rebuke.
4)the words of the Mishnah and Talmud lack the formula, "And the Lord spoke unto Moses saying," and "Thus saith the Lord".
The mishna and talmud also contain no phrase like "you may not kindle a fire on the Sabbath" or "you must wear tefillin" or a direct instruction to perform virtually any mitzvah! Look for yourself! There are only specific technical details for the practical observance of the commandments (both Divine and rabbinic). The rest is implied and is understood as obvious by its readers.
In the traffic code of a modern society, are there many phrases like "the Government of the State of New Jersey requires that its citizens and visitors should not drive in excess of..." or "the philosophical tenets of a social contract imply that our residents must come to a full stop at all stop signs..."? Of course not, the origin and authority of the law are understood by everyone. The mishna is no different (except that its stated goal is to be brief and succinct - a purpose enhanced by leaving out obvious and intuitive material).
Also, the formula you mention above ("and the Lord spoke unto..."), when found in the written Torah, is used to convey the message that these exact words were communicated. The mishna, however, is not comprised of G-d's exact words.
5)the Mishnah and Talmud are full of the opinions of rabbis who disagree with each other on almost every issue. Whenever there are such disagreements the rabbis claim that both opinions are the words of God?!
Indeed. Actually, this is a hugely complex issue and I would advise you to read the fifth volume of Rabbi S.R. Hirsch's Collected Writings for a better picture. I, however, will try to give you a brief answer. We must first distinguish between different types of Talmudic passage: halacha and aggada (medrash). First we'll discuss the halacha.
When Rabbi Yehuda Hanasi convened the rabbis to commit the mishna to writing (end of 2nd Century, ce), he gathered all the sages together to present before him every known halachic tradition. Rabbi Yehuda then, using the technical structures that had always existed to decide between dissenting opinions, selected as authoritative those meeting the Torah's criteria (the Torah itself already allowed for the possibility of dissension and provided a mechanism for resolution - see Deut. 17; 11 and Ex. 23; 2).
So you might ask "why, then, are all these alternate opinions recorded in the mishna? Shouldn't only the halachic opinion have been included?" The mishna itself asks that very question (Edioth 1; 5):
"And why (does a mishna) mention the words of an individual (i.e. the non-halachic opinion) in the face of a plurality since the halacha always follows the plurality? So that if one (later) court should consider (independently) the ideas of the individual (they will be aware that that idea has already been suggested and rejected)..."