The American colonists famously rebelled against taxation without representation; did the British consider acquiescing on this point before things escalated to violence?

This previous answer of mine should give background to your question. To recap that answer: The Americans were less concerned about the taxes themselves and more concerned with the structure of the government in regards to tax legislation. The colonies believed that their colonial charters (pre-cursors to the state constitutions) made their local legislatures co-equal branches with Parliament, at least on tax matters. It was in these local legislatures where the colonies had their right to "representation," hence the slogan. Parliament could not unilaterally enact tax legislation upon the colonies without getting the respective colonial legislatures to agree to the tax. Essentially, they believed each colonial legislature had veto power over any tax legislation passed by British Parliament at Westminster. Before the 1760s, Parliament had almost never tried to impose such taxes unilaterally, and now they were.

(It had actually happened twice before, through indirect taxes: the Plantation Duty of 1673, and the Molasses Act of 1733. In both cases, the taxes were met with protest, and largely went unpaid. They were both taxes paid on imports, so the taxed goods would be regularly smuggled past British tax agents at seaports, or else the agents would be bribed to ignore the taxable goods.)

Further, the American colonies never really desired seats in British Parliament. They just wanted their local representation to have the power of taxation. In fact, seats in Parliament would kind of undermine their legal argument if they accepted them, because it would be a tacit admission that the language in their colonial charters did not make them co-equal on tax policy, as they had been arguing. But that's a bit of a moot point, since Parliament never really made such an offer.

More directly to your question, then: Did Parliament ever consider acquiescing to the colonial view of the law, that their local colonial legislatures were co-equal and had veto power over tax legislation in the colonies?

The answer is no, not really, at least, not until it was too late. First and foremost, this is exemplified by the fact that Parliament kept passing new tax laws without colonial input, even after being met with non-compliance by the colonies, and petitions by the colonies about why they considered each of these tax laws illegal. The Sugar Act (1763) was followed by the Stamp Act (1765) which was followed by the Townshend Acts (1767) which was followed by the Tea Act (1773).

After the Stamp Act was passed, the Massachusetts legislature met, and the royally-appointed governor of the colony, Sir Francis Bernard, addressed the legislature. While he was not particularly thrilled with the Stamp Act, he was nonetheless the British government official tasked with enforcing it. Governor Bernard consulted with the British ministry (i.e. the administration of Prime Minister Rockingham), after which he said in his address to the Massachusetts legislature:

"I only have to say that it is an act of the Parliament of Great Britain, and as such ought to be obeyed by the subjects of Great Britain. And I trust that the supremacy of that Parliament over all the members of their far and wide and diffused empire, never was and never will be denied within these walls. The right of the Parliament of Great Britain to make laws for the American colonies, however it has been controverted in America, remains indisputable at Westminster."

The 1765 Stamp Act inspired a major crisis, and the colonies formed the Stamp Act Congress shortly after to protest what they saw as an illegal law. Parliament ultimately backed down and repealed the act, but did so in conjunction with passing the Declaratory Act of 1766. This act declared that Parliament "had hath, and of right ought to have, full power and authority to make laws and statutes of sufficient force and validity to bind the colonies and people of America...in all cases whatsoever." In other words, Parliament might be repealing the Stamp Act, but they still reserved the right to enact taxes on the colonies at the Parliamentary level.

The Intolerable Acts were passed in 1774, which, in part, punished Massachusetts for non-compliance with the Tea Act, and the Boston Tea Party. After this, there arose a real threat of revolution in Massachusetts and elsewhere. To fend this off, Parliament passed the Conciliatory Resolution in February 1775, which agreed to recognize the colonial legislatures as co-equal on taxes that would go to the "common defence" (i.e. taxes that went to military matters), but that Parliament would still reserve the right to unilaterally raise taxes "for the regulation of commerce".

A committee of the Continental Congress reviewed the Conciliatory Resolution, and wrote a report calling it "inadequate" since it didn't address taxation as a whole. The report also said that the Resolution "seems also to have been calculated more particularly to lull" the colonies "into fatal security" than to be a good faith effort by Parliament, fearing that Parliament would start imposing "common defence" taxes just as soon as it was convenient for them to do so. The Conciliatory Resolution was never brought to the floor of the full Congress for consideration.

In July 1775, after the Battle of Lexington and Concord, the Continental Congress sent the Olive Branch Petition to the King, in an effort to prevent outright war, by suggesting the two sides negotiate tax and trade regulations that would be accepted by both sides. Reportedly, the King didn't even read it, already considering the Congress to be traitors, and made this his official view when he issued the Proclamation of Rebellion the following month.

In July 1776, Congress issued the Declaration of Independence. This was followed directly by the Battle of Long Island, the first major battle conducted under the leadership of the British armed forces against the Continental Army. After the battle, but before the British had captured New York City, the two sides conducted the Staten Island Peace Conference, where Admiral Lord Howe was authorized to offer the Americans a peace deal on behalf of Parliament and the King. Great Britain would declare a cease-fire and offer pardons to all rebels if the Americans would agree to the Conciliatory Resolution, and would agree to dissolve the Continental Congress in exchange for the re-establishment of all the colonial legislatures (Parliament had dissolved the Massachusetts assembly as part of the Intolerable Acts). Parliament would agree to negotiate equitable resolutions to all other outstanding grievances as well, aside from the Quebec Act, which would stay in effect.

The Americans rejected the proposal. Aside from all the other disagreements the two sides had by then, the Conciliatory Resolution was still only a half-measure on tax policy. Again, the Conciliatory Resolution only provided for recognition of colonial power over military-related taxes, while commerce-related taxes would remain a power of Parliament. The Americans maintained their argument that it was their British constitutional right to have authority over all tax matters.

Finally, the British government did acquiesce the point, with the Taxation of Colonies Act of 1778, which declared that Parliament would not impose any tax laws on the American colonies whatsoever. Of course, this was well after the war was underway, and after British General John Burgoyne had been defeated at the Second Battle of Saratoga in 1777, often considered the turning point of the war in the Americans' favor. It was the point which emboldened France to join the American side, which in turn strengthened the possibility of an American victory. So the 1778 act was very much "too little, too late" as it was an effort to end the war by keeping the colonies under British control when there was a solid possibility that Britain would lose the war and the colonies altogether.

But even that law contained no guarantee that it could not be overturned by future Parliaments. Regardless, Congress didn't even consider it. The war continued, and eventually the Americans won, which, of course, meant the Americans gained local control over all tax matters.

Were the American colonists asking for extraordinary rights for an overseas territory, or were they being singled out?

This part of your question would be better addressed by someone else, since I don't know much about Great Britain's other overseas territories at the time. British control over the Canadian provinces was too new to have much of a history. Local legislatures there under the crown were only formed after the French and Indian War, so there wasn't any precedent there for taxation. In the West Indies, the most I can say is that the taxes to be paid there under the Plantation Duty of 1673 and the Molasses Act of 1733 were as ignored there as anywhere else. And then when Parliament passed the Taxation of Colonies Act of 1778, it applied to the West Indies as well as to the Thirteen Colonies and Canada. So the Thirteen Colonies weren't necessarily being singled out on taxation policy, but their view of Parliament's rights to tax colonial citizens didn't seem to differ entirely from the viewpoint of some of the other overseas territories. But Parliament did target the Thirteen Colonies, and in particular Massachusetts, in other ways, particularly with the Intolerable Acts, which was largely the impetus for pushing the Thirteen Colonies over the top toward revolution.

EDIT: Added info about the Declaratory Act.