You do not need a synset file unless you're doing ImageNet classification - this provides hierarchic classes information so you can give your network a partial credit: it might have failed to guess the exact breed of dog in the picture, but it did get the "animal", "mammal", and "dog" subclasses correctly. No need for that level of complexity if you're doing simple tasks like OCR.
LMDB does not contain label information aside from raw indices, that is: your examples (images) are paired with labels (integers) but no label descriptions are packed with them. So you know image a shows an object of class 4 and image b - class 17, but you don't know what those classes are until you look into a label file, which provides those descriptions, listing label names so that you can associate them with indices. You might get away without them in this particular case when you're classifying digits and class 0 corresponds to string "0" and class 1 to "1" etc. but if you want to know if your network found a dog or a cat in an image you need it.