Daniel 1 – Skeptic's Annotated Bible answered

To solve a case like this, it helps if we compare apples with apples.
First of all the exact date is 607 BC according to Bishop Ussher, not 606 BC. This was indeed the first year of the reign of Nebuchadnezzar if we just count as the Bible counts. And the first year of Jehoiakim starts in 610 BC. As our years don't fit nicely into the reigns of the kings (different starting month), the third year is 607 BC.
Now to the supposed bad history. The author of the SAB quotes here Farrell Till as saying that the first invasion was at 597 BC, which would be the last year of Jehoiakim. Farrell Till concludes this from Babylonian records. And Farrell Till makes a second claim, namely that Nebuchadnezzar became king in 605 BC, two years after the Bible says he besieged Jerusalem. All these dates can be a bit confusing, so let me give a time line below (all dates in BC), and according to Ussher's chronology:
610: Jehoiakim is made king by Pharaoh Nechoh, see 2 Kg. 23:34.
608: Nebuchadnezzar is made viceroy.
607: battle of Carchemish, where Egypt was defeated (fourth year of Jehoiakim), see Jer. 46:2.
607: first siege of Jerusalem by Nebuchadnezzar; Daniel and his friends taken and brought to Babylon, start of the 70 years of captivity.
605: death of Nabopolassar, father of Nebuchadnezzar.
604: Jehoiakim rebels (2 Kg. 24:1), it takes Nebuchadnezzar a few years to be able to respond.
600: Nebuchadnezzar invades Judah, see 2 Kg. 24:2.
599: second siege of Jerusalem; Jehoiakim's son Jeconiah, also Jehoiachin, was taken to Babylon (Jer. 27:20), see also 2 Kg. 24:12.
589: last siege of Jerusalem by Nebuchadnezzar.
588: destruction of the first temple.
Note that secular history gives 586 BC as the date that Solomon's temple was destroyed. And note that Bishop Ussher's starts the reign of Nebuchadnezzar two years earlier than secular history.
We know the relationship between the reign of Nebuchadnezzar and Jehoiakim, and this allows us to match their reigns. According to Jer. 25:1 the fourth year of Jehoiakim falls within the first year of Nebuchadnezzar. From this verse we know that Nebuchadnezzar's reign began during the third year of Jehoiakim.
So this allows us to solve the second “contradiction” claimed by Mr. Till: he compares Ussher's chronology as used in many Bible dictionaries with dates used by secular historians, and so obtains his contradiction that Nebuchadnezzar wasn't even king. In secular history, Jehoiakim begins his reign in 609, and Nebuchadnezzar begins his reign at 605. But secular dates are simply wrong, as the Bible does not contradict itself and internally aligns the reigns of these two kings perfectly and consistently. It's only when we start using dates obtained by some method outside the Bible that we can obtain a contradiction.
It is correct that Nebuchadnezzar's father died in 605 BC. When that happened Nebuchadnezzar became sole ruler. But from the Bible we know that as viceroy he reigned already 2 years before that. It is the normal procedure in the Bible to count the reign of a king from the first year that he became a viceroy.
This also explains why he was in Jerusalem in his first year. If he had become king, he probably would not have ventured too far outside his realm in the first year, but would be establishing his power base in his home country and close at home. But because his father still reigned, he was able to make fame for himself by venturing abroad to obtain glory and loot.
There is one issue that might seem contradictory: from Jer. 46:2 we know that the battle of Carchemish was in the fourth year of Jehoiakim, while this verse seems to say that in the third year of Jehoiakim we find Nebuchadnezzar before Jerusalem, which happened after the battle of Carchemish. But it we read this verse correctly, it says that Nebuchadnezzar came in the third year, that is, he left with his army at the end of the third year of Jehoiakim. From Jer. 46:2 we know when he arrived: in the beginning of the fourth year of Jehoiakim.
A different resolution is offered by Mark S. Haughwout, who claims:

This apparent discrepancy with Daniel’s account above is actually a cultural difference of dating systems. Jeremiah, living in the land of Israel, naturally uses the Israeli dating system, which would place Jehoiakim's fourth year in 604 BC. Daniel, using the Babylonian system, places Jehoiakim's third year in 604 BC. The Babylonians considered a king’s first year to start on the first New Years day in his reign.

Having established the date of the siege, let us have a look at the claim that the first siege of Jerusalem was only in 597. This is unlikely. If we look at the Biblical record, we see that Pharaoh Nechoh was expanding his kingdom to the North and East, something the Babylonian Chronicles also record. On his way back from Harran, Pharaoh Nechoh disposed the king in Jerusalem and installed Jehoiakim, 2 Kg. 23:34. Farrell Till would make us believe that it took more then 10 years for Nebuchadnezzar to respond to this act of Pharaoh Nechoh. But this act of Pharaoh Nechoh is actually the reason we see Nebuchadnezzar before Jerusalem! In his first year as viceroy, he begins with his campaign to defeat Pharaoh, which he did convincingly at Carchemish. Afterwards he marches south to attack Pharaoh's allies and dispose his vassals. And that is why we find him before Jerusalem at this time and Daniel and his friends were taken to Babylon. His venture south ends with the full retreat of Pharaoh, see 2 Kg. 24:7.
So the first siege was in 607, the second siege was in 599 (not 597 as secular history has it), and the third and last in 589 BC.

Daniel's issue with the wine was not that it contained alcohol. Alcohol was only forbidden for those who had taken the vow of a Nazarite, see Num. 6:3. The issue was that the wine was an oblation to idols.
Daniel's issue with the meat was not that it was meat. But the issue with the meat was that it might wholly or partially contain meat he was forbidden to eat, such as pork, or that the animals might not have been killed as per Lev. 7:26, i.e. that the blood was drained.
In chapter 10:3 we see that Daniel used wine and meat in addition to bread when he was able to provide himself with kosher food.

Why does the author of the SAB laugh? Because it wasn't true? Why then were Daniel and his three friends given such prominent positions? This was not boasting. These were the king's own words.