It gets a little more complicated, because barring is on the Z sex chromosome, and so is chocolate (although I'm told they re-combine enough that they don't act linked to each other, even though both are clearly sexlinked.)
So I would expect:
--every F1 hen passes barring to her sons. She has no chocolate to give them, so her sons will never show chocolate. She gives a W chromosome to her daughters, so she has no effect on whether they will have barring or chocolate.
--F1 males will pass barring to half of their chicks, and not-barred to the other half of their chicks. This means daughters can be barred or not-barred, and sons can be double-barred or single-barred (because each son gets barring from his mother.) Likewise, F1 males will pass chocolate to half of their chicks, and not-chocolate to the other half. So daughters may be chocolate or not, sons will not show chocolate but have a 50% chance of carrying it.
That works out to the following options for barring and chocolate:
males, 25% each of single barred carrying chocolate, single barred without chocolate, double barred carrying chocolate, double barred without chocolate.
females, 25% each of barred, chocolate barred, chocolate not-barred, neither chocolate nor barred.
For blue, yes you should get 25% splash, 50% blue, 25% black (not-blue). That applies to each of the above groups, so they'll break down like this:
single barred males --> 25% splash single barred, 50% blue single barred, 25% black single barred
chocolate females --> 25% splash/chocolate, 50% mauve, 25% chocolate
(etc. for each other group-- I'm not going to list every one of them.)
For lavender, since the F1s all carry it, it will also show up in 25% of chicks, be carried by 50% of chicks, and not be present at all in 25% of chicks. So each group above will have that same ratio: 25% of females are chocolate, 25% of them are not-blue, 25% of those are lavender. Or in fractions, 1/4 of females are chocolate, 1/16 of females are chocolate and not-blue, 1/64 of females are chocolate and not-blue and lavender. 2/64 of females are chocolate, not-blue, carry lavender without showing it.  1/64 of females are chocolate & splash & lavender. etc.
In practice, there are so many colors & combinations you won't see them all unless you hatch a very large number of chicks.
This is one where it's easier to use the chicken calculator:
http://kippenjungle.nl/kruising.html
For each gender of the F1, put in:
E/E (pure for Extended Black)
B/b+ male or B/- female (each has one copy of the barring gene)
Choc+/choc male or Choc+/- female (only the males carry chocolate)
Bl/bl+ (each is blue)
Lav+/lav (each carries the lavender gene)
Leave all the other genes on the default settings (all are marked with + to indicate the wild-type form.) Most of them are correct, and a few may be wrong but won't actually affect what you see of the offspring (like gold/silver, that doesn't show in all-black chickens.)
Click the button for "calculate crossing"
You will get a very long list of images with the genes listed, including what they show and what they carry, and it will tell how likely each one is to occur (many are about 1/128 likelihood, which accounts for 1/2 of chicks being each gender and 1/64 of that gender having a certain color.)
You can use the calculator on your other crossings, too. The genes I listed are the only ones you should need to mess with for any of them.
Close.
Barring will appear in half the chicks, but not the other half, because the F1 father only has 1 barring gene, not two, and the mother doesn't have any.
Mauve and mauve splash should show up half the chicks (the ones that inherit chocolate from the F1 male.)
The other half of the chicks will inherit not-chocolate from him, so they will be blue or splash.
Half the chicks will carry lavender (from the F1 male), but none will show it.
So about equal numbers of mauve, mauve barred, mauve splash, and mauve splash barred.
Yes, you have that right.
Crossing the barred F1 daughters back to their barred father will give 100% true-breeding barred chickens (single-barred females and double-barred males.)
Half of the sons will carry the gene for chocolate, but the other sons will not, and none of the daughters will.
Close! You have lots of genes involved, so it gets pretty complicated!
The backcross of daughters to father is genetically the simplest, and the straight F2 is the most complex by far.