In the Harry Potter universe there are three kinds of wandcore widely used: phoenix tail feathers, dragon heartstrings and unicorn hairs. Do these variations in wand cores affect the wand's ability? For example, does a dragon heartstring core allow a wizard/witch to perform flamboyant spells better?

There are also occasionally special types of elements used in wands. For example, Slytherin's wand contained a fragment of basilisk horn. Did this give any special ability to his wand?

The Wand chooses the Wizard and it's apparent that certain kinds of woods are better fit for particular types of magic.

From Pottermore

Acaica only works for the very best of magic users.

A very unusual wand wood, which I have found creates tricky wands that often refuse to produce magic for any but their owner, and also withhold their best effects from all but those most gifted. This sensitivity renders them difficult to place, and I keep only a small stock for those witches or wizards of sufficient subtlety

Alder is best for non-verbal spells

Of all wand types, alder is best suited to non-verbal spell work, whence comes its reputation for being suitable only for the most advanced witches and wizards.

Ebony is best for transfiguration

This jet-black wand wood has an impressive appearance and reputation, being highly suited to all manner of combative magic, and to Transfiguration.

Same with Fir

Fir wands are particularly suited to Transfiguration, and favour owners of focused, strong-minded and, occasionally, intimidating demeanour.

Now regarding Wand Cores, there is similar information

Unicorn hair cores are bad for Dark Art magic:

Wands with unicorn cores are generally the most difficult to turn to the Dark Arts. They are the most faithful of all wands, and usually remain strongly attached to their first owner, irrespective of whether he or she was an accomplished witch or wizard.

Consequently, you would be inclined to conclude that they are good for "light" spells like the Patronus Charm and the Fidelius Charm(which is based on faith and kinship).

As you said, Dragon heartstring is good for flamboyant spells and dark arts

As a rule, dragon heartstrings produce wands with the most power, and which are capable of the most flamboyant spells.

The dragon wand tends to be easiest to turn to the Dark Arts, though it will not incline that way of its own accord. It is also the most prone of the three cores to accidents, being somewhat temperamental.

Phoenix feather wands are best for a large range of magical arts.

This is the rarest core type. Phoenix feathers are capable of the greatest range of magic, though they may take longer than either unicorn or dragon cores to reveal this.

Salazar Slytherin's Wand

His wand had the most affinity to himself and his descendants. The only special ability I could find was it's ability to "sleep" and thus ensuring it's true allegiance is held by a member of the Slytherin family who knows the sleep and awake parseltongue words.

However, Isolt had never known that she held in her hands the wand of Salazar Slytherin. While tracking down her niece, Gormlaith sent a powerful curse containing Isolt and her husband’s names towards the house, which forced them into an enchanted slumber. Gormlaith uttered a single sibilant word in Parseltongue, the wand quivered and became inactive. When Isolt and her husband was awakened out of their enchantment because of the cries of their twin daughters, Isolt tried to stop her aunt together with her adoptive sons, but it was useless with a deactivated wand.

Yes, wandcores do affect the wands abilities.

Unicorn Hair Produces the most consistent magic.

Unicorn hair generally produces the most consistent magic, and is least subject to fluctuations and blockages. Wands with unicorn cores are generally the most difficult to turn to the Dark Arts. They are the most faithful of all wands, and usually remain strongly attached to their first owner, irrespective of whether he or she was an accomplished witch or wizard.

Minor disadvantages of unicorn hair are that they do not make the most powerful wands (although the wand wood may compensate) and that they are prone to melancholy if seriously mishandled, meaning that the hair may ‘die’ and need replacing.
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Dragon Heartstring create the most powerful wands

As a rule, dragon heartstrings produce wands with the most power, and which are capable of the most flamboyant spells. Dragon wands tend to learn more quickly than other types. While they can change allegiance if won from their original master, they always bond strongly with the current owner.

The dragon wand tends to be easiest to turn to the Dark Arts, though it will not incline that way of its own accord. It is also the most prone of the three cores to accidents, being somewhat temperamental.
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Phoenix Wands possess the greatest range of magic possible.

This is the rarest core type. Phoenix feathers are capable of the greatest range of magic, though they may take longer than either unicorn or dragon cores to reveal this. They show the most initiative, sometimes acting of their own accord, a quality that many witches and wizards dislike.

Phoenix feather wands are always the pickiest when it comes to potential owners, for the creature from which they are taken is one of the most independent and detached in the world. These wands are the hardest to tame and to personalise, and their allegiance is usually hard won.
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As well as this Woods also affect the abilities of the wand .

Readers should bear in mind that each wand is the composite of its wood, its core and the experience and nature of its owner; that tendencies of each may counterbalance or outweigh the other; so this can only be a very general overview of an immensely complex subject.
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