To answer your question “Does he stand a good chance if I go to appeal?” it rather depends on the strength of the case you present. Ideally you need alternative evidence of high academic ability, and some extenuating circumstances to explain the underperformance.
For further information, have a look at the appeals section of this website:
http://www.elevenplusexams.co.uk/11plus ... nswers.php
If level 5s have been predicted, it does sound as if George is a very good pupil, but reasoning tests do not measure exactly the same thing as SATs.
It would be worth asking your headteacher for advice about an appeal. After all, if you decide to go ahead, you will want his/her support.
Some grammar schools in Lincolnshire, such as Queen Elizabeth’s, Gainsborough, are under local authority control. However, if the school you are interested in is a foundation grammar school and its own admission authority, then your appeal would be against the school.
As for the common application form, as far as I know Lincolnshire (but not necessarily the foundation grammar schools) operates an “equal preference” system, so if the alternative schools you have in mind are under local authority control, I think it would make sense to put the grammar school at the top of your list because that will not disadvantage your other choices.
I'm not an expert on Lincolnshire, but I hope this will be of some help.