An integration should be possible but it’ll be WebExpenses or a third party developer who is likely to set it up, rather than Monzo.
This isn’t possible yet but once the API is ready, it will be made open so that once you’ve authorized an application, it will be able to see your transaction history, transaction categories, attached receipts, deposits (for reimbursements) etc. & then their app can process all of that data.
The only items on the roadmap that are specifically for the API are in the ‘long term’ list but Integrations with TFL, Nectar etc. (which will presumably require a working API), are due within the next 6-9 months, once the bank is launched. So hopefully the API will be ready for this sort of thing, which only requires read access, by then.