I know nothing about Gladys, not even her surname, but I like the nickname that Gladys uses for Kathleen, even though Kathleen apparently didn't, since she never used it for herself.
Dear ‘Gasolene,’ How are you getting on now? Kate told me you were 2nd in class. Do you think Mac. has forgotten our albums? I like being at school very much. I’m going to Centre next Monday. Will you write to me soon? With love from Gladys
...so maybe that’s who this was. But I doubt that I'll ever know for sure.
3. 4 Wellesley Road, Wanstead, Essex - that's where Kathleen was born, and where the Farmers lived for many years.
4. Gasolene: this isn't a mis-spelling, but rather the original spelling, which we don't use anymore. From Quora I got:
USA — Gasoline — originally ‘gasolene’ (1863), possibly influenced by the trademark ‘Gazeline’ — abbreviated to ‘gas’
In the UK, they say petrol instead of gas, but maybe the usage was more fluid back then when cars were a new thing, or maybe somebody heard that petrol was called gasolene in the US, and happily applied the word to Kathleen.