Logs need ventilation so most people use a log store something like this or larger. Easy to buy or build.

I doubt that sort of portable/mobile structure would be subject to any legal issues if built entirely on your land.
You may well be liable for any damage to her property but they would have to prove your building caused it. I believe they also have to notify you of the problem and you only become liable if you then do nothing about it...but best check that on the law forums. Your house insurance may also cover you.
If you are building something more permanant in brick or which needs foundations then check...
a) If you need planning permission..
http://www.planningportal.gov.uk/england/public/buildingwork/projects/workcommon...b) If you need Building Control Approval...
http://www.planningportal.gov.uk/england/public/buildingwork/projects/workcommon...Probably not needed if less than 15sqm but report back if you plan to put electricity/lighting in.
c) If the Party Wall Act applies...
http://www.communities.gov.uk/documents/planningandbuilding/pdf/133214.pdfUnlikely to be an issue if built entirely on your land and the foundations aren't deeper than the neighbours foundations. If you want to fix it to the neighbours house you will almost certainly have to comply with the act.
d) Covenants in the deeds. There might be clauses that prevent you building close to the neighbours house or which require you to allow them access to their wall for maintenance. I guess some people may even have a covenant that prevents them burning wood or non-smokeless coal.
You can normally build right upto the boundary of your property but if there is an adjacent building it's best to either build onto the adjacent property with no gap or leave a gap large enough for a person to get down to rake out leaf debris etc. Something like a small six inch wide gap is the worse option as it allows stuff in but makes it impossible to remove. It can build up until the damp proof course is bridged in either property.