Paying guest loses a fringe (6)
Vermin back in best Armenian families (4)
Mates in turmoil giving off vapour (5)
Tune on the radio ? (3)
WHERE THE EXECUTIONER'S AXE FALLS ? (5,7)
Food comes back as American time period (7)
Born in Little Rock, Arkansas - grew up in Colorado ! (7)
Darn fire is rubbish - more heat than light ! (5-3)
On the other hand the Arkansas clue breaks at least two rules. Firstly the definition is not "Little Rock, Arkansas grew up" but simply "Little Rock grew up". And then, if we're being pedantic (which we are), it really ought to be "Little Rock grown up". The second rule is that the definition should never come in the middle of the clue, always at the start or the end.
Everyone has their own stance on what is acceptable and what is not. I consider myself a moderate compared to Ximineans like Don Manley (whose Crossword Manual is well worth a read). However, you might consider me onerously fascist for these opinions.
I expect now for you to reveal that it was taken from a Times puzzle...
With regards the capital one : The executioner's chopping board was known as the block capital (admittedly perhaps only poetically) as it was the block on which capital punishment was "executed".
And I never attach my box naked to the net.
[all] Ummm, little miss untechnological strikes again. How do I do hyperlinks?
Meanwhile, the cryptics. I can recommend two books. One is 'Cryptic Crosswords and How to Solve Them' (published I think by Chambers - you can search Amazon for example). The other is Don Manley's Crossword Manual, linked to above. The former is better for jumping in. Basically it lists the devices, and then gives some graded puzzles, with the amount of help diminishing as the book progresses. My main gripe is that the authors do like (a) to show off how clever they are; and (b) literary allusions rather than give you practical advice to clue-solving. Manley's book is much more thorough, but as he treats things more from a historical perspective, you'll find yourself dealing with Victorian mindbenders before getting into the cryptics proper. However, this book does offer an insight into the mind of a setter, which is invaluable.
One thing I dislike is microclassification of clues. Since pretty much every clue involves a combination of devices the only classification is: