Can you carve a wand after it is created?

Question

As can be seen by some of the Death Eaters' wands, many have shapes and objects carved into them to indicate an affiliation or as decoration:

Alecto Carrow's Wand:

Amycus Carrow's Wand:

Various Death Eater Wands:

Obviously, you can attach things to wands, as evidenced by Lucius Malfoy's wand, his wife Narcissa’s, and many other examples:

However, many of the other wands appear to either be carved or that was how they were originally produced. So what influences the shape of the wand? Can the owner of a wand modify/carve it after acquisition?

Answer 1

Probably Not, These Features Were Added In The Movies

In the books, there is no mention of these type of embellishments. There are only two features: the type of wood and the core (see here and here):

WANDS - Made from various kinds of wood, containing a core from substances such as dragon heartstring, unicorn hair or phoenix feather. (Pottermore factsheet)

The absence of peculiar features makes sense given that the owner is not known when the wandmaker creates the wand. Although the wand-making art is never explored in detail, I would argue that later changes made by the owner would count as damages. First, there is no such thing as a part-time wandmaker. Second, wands are not entirely inanimate. JKR said

Essentially, I see wands as being quasi-sentient, you know? ... They're not exactly animate but they're close to it.

In the movies, they made a diverging artistic decision:

Each wand in the Harry Potter films was developed with its fictional owner in mind.

The wands in the movies were meant to mirror the characters' physical appearance and/or personality traits - and probably to make the merchandise more attractive to fans. In fact, embellishments were not confined to the Death Eaters, as you can see on the dedicated Pottermore page.

Alecto Carrow's wand is a good example of this book/film inconsistency since the skull wand is different from the one in "Harry Potter: The Wand Collection", which has no character-specific embellishments (see here).

Also on the character-specific Pottermore pages, there is no mention of embellishments of their wands. As in the case of Lucius Malfoy, you can only read:

WAND Eighteen inches, elm, dragon heartstring. After the Battle of the Seven Potters he began using an unknown wand.

Answer 2

Wands can be carved - but it’s unclear if also after they’re made.

There are example of carved wands outside of the movies (in the movies there are several unique styles of wand) - James Steward and Isolt Sayre give their adoptive son Chadwick a wand that’s described as being finely carved - so carved wands do exist outside of the movies. In addition, the carving was most likely done by James, who was a Muggle, so had no magical skill.

The Horned Serpent was waiting there for her. It raised its head exactly as it had done in her dream, she took part of its horn, thanked it, then returned to the house and woke James, whose skill with stone and wood had already beautified the family cottage.

When Chadwick woke next day, it was to find a finely carved wand of prickly ash enclosing the horn of the serpent. Isolt and James had succeeded in creating a wand of exceptional power.
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Shikoba Wolfe, an American wandmaker, was famous for making intricately carved wands.

Shikoba Wolfe, who was of Choctaw descent, was primarily famous for intricately carved wands containing Thunderbird tail feathers (the Thunderbird is a magical American bird closely related to the phoenix).
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These wands must work, since he’s considered one of the four great wandmakers in America, so wands can be carved and still functional. However, it remains unclear if they can be carved after their creation. Chadwick’s wand was carved when it was being made, not after, and it’s not said when in the process of making wands Shikoba Wolfe carves them. Since it is possible for wands to be carved, it seems like it should also then be possible for a wizard to have their new wand carved in a custom way to make it look distinctive or suit them better.

Answer 3

We know the wands are quasi-sentient in the HP universe. They possess a certain amount of understanding of who their owner is, they may or may not be loyal, they may or may not be particularly willing to cast a specific spell, they are said to embark on a learning quest together with their owner.

On the other hand, in the HP universe, magic can affect the appearance of a person using it (or affected by it). He-who-must-not-be-named is the most famous example: his use of Dark magic changed his appearance so much nobody could recognize him as Tom Riddle anymore. Another example is Lupin saying to Sirius: "finally the flesh reflects the madness within".

It stands to reason, therefore, that the wand itself, as it gathers the experience and learns and bands together with its master -- will eventually morph and take on the shape that will reflect its master's character.