I'm sure that's not true - there are hundreds of children crossing the Birmingham/ Solihull border (in both directions!). If you live in Birmingham you will presumably not be in the catchment area for any Solihull schools but should be able to apply in theory to any school in the country. I think you need to list the Solihull schools on your Birmingham preference form (take great care with the ordering - you will automatically be allocated to the highest preference school for which you meet the criteria). What I don't know is whether you need to fill in another form for Solihull as well
You can apply to any school anywhere you want, but if, as in the case of the Solihull schools, their admission criterion is distance between home and school you would be wasting those choices on your LEA form unless you are sure you live within their catchment area. Unrealistic school choices are the commonest reason why parents end up not being offered any of their preferences. Unless the schools are Foundation ones you won't need to fill in anything other than your Birmingham LEA preference form. You need to make sure that you have at least a fighting chance of matching the admissions criteria of the schools you apply to, though, otherwise your child will be allocated to the nearest school which still has vacancies, whether it was on your list or not.