Is it possible to have 2 applications write to the same log file using log4net?
They can but if one application is writing the the file the other application will most likely experience an error if it needs to write to the log as well due to the fact that the first application will be holding the file open for writing. It is always best to have dedicated logging sources for your applications - if you need to share a log, use a database as it is designed to handle concurrent writes.
This is one of those things that will work really well on your machine when you are developing since you are not likely to create enough concurrent writes to the log file to notice any problems. Once your application begins to experience more load the problem will begin to show itself and at that point it may manifest itself in strange ways. I would definitely try another solution.
It depends on the FileAppender's LockingModel. If it is ExclusiveLock then another process cannot open the file for writing. The alternative is MinimalLock, but it is not meant for this purpose. It is intended for allowing another process to move or delete the file.