Whilst this puzzle can be solved without knowing the identities of the four caricatured celebrities, it does help to realise who they are, as they have each been chosen for a reason...
Slash, the guitarist from Guns 'N' Roses (top hat, long hair);
Rod Stewart, the singer (messy blonde mop);
David Bowie, the musician (red hair, face paint);
and Barry White, the R&B singer-songwriter (in the flat-cap)*.

These identities are relevant, as this puzzle involves...
...forming the shapes of letters from their component straight and curved lines.
Specifically, a letter is indicated by a combination of:
- diagonal lines (or 'slashes', depicted by Slash, oriented in one of two ways depending on the lean);
- vertical lines (or 'rods', depicted by Rod Stewart);
- curved sections (or 'bows', depicted by David Bowie);
- horizontal sections (or 'bars', depicted by Barry White).
So, for example, the 'G' in 'GEORGE' is made with a curved section, a short vertical rod, and a short horizontal bar - a Bowie, a Rod, and a Barry. The 'R' is made with a slash to the right, a curved section, and a tall vertical line - a right-facing Slash, a Bowie, and a Rod.
All letters can be formed in this way, but beware - not all are uniquely identified. An 'O' and an 'S', for instance, can both be indicated by two Bowies - two semicircle arcs connected together differently. This is particularly important for the first question...
Question A: The name here is:
SHANIA (as in Shania Twain, the musician).
Two Bowies gives us 'S' (see note in spoiler above);
A Barry and two Rods gives us - | | which can be arranged into an 'H';
Two opposing Slashes and a Barry gives us / - \ which can form an 'A';
A rightward Slash and two Rods gives us \ | | which can form an 'N';
A single Rod is an 'I' (we know this from the examples);
A second A to finish.
Question B: The band is:
WEEZER! Note here that the letters of the band have just been decomposed into their constituent parts and arranged by grouping the four different caricatures together. To break down the 'QUEEN' example, we need:
Q = Two Bowies and a rightward Slash;
U = Two Rods and a Bowie;
E = Three Barries and a Rod;
E = Three Barries and a Rod;
N = Two Rods and a rightward Slash.
Together this forms the requisite 2 rightward Slashes, 6 Barries, 3 Bowies, and 6 Rods.
How to solve the 6-letter band name? Note that there are 22 faces in all. Since all letters can be made from a maximum of 4 segments, the most you can have for a 6-letter word is 24 faces. That means at least 4 of the letters must use 4 faces.
Since an 'E' requires a Rod and 3 Barries, and there are an awful lot of Barries used here, let's max out on E's and use 3 of them, and also make up a 'W' using 4 Slashes (like so: \/\/). We have the requisite faces left over to form an 'R' (Bowie+Rod+Slash) and a 'Z' (last 2 Barries and the other kind of Slash), which ultimately confirms my choice of WEEZER!
* Thanks @hexomino in comments for spotting the exact identity of one of the faces (I had previously been reliant on their flat-cap in my write-up up until then...