The ERA archive description document, Berrisford et al (2011) "The ERA-Interim archive Version 2.0", provides additional information about what is available. It's less explicit on the why, but we can read between the lines a bit.
The archive consists of re-analyses of the system state at 00, 06, 12, and 18 UTC and re-forecasts out to 10 days initialised from those 00 and 12 UTC states. Table 9 of that document lists the variables (including the ones that you're interested in) that are stored as accumulations since the start of the re-forecast. They are stored every 3 hours for the first day of the forecast (steps 3, 6, 9, 12, 15, 18, 21, 24) less frequently thereafter (every 6 or 12 hours).
The variables are not stored for step 0 because their accumulations would be exactly zero.
Regarding using ERA Interim for model forcing, you might consider what Balsamo et al (2015) do for ERA Land forcing. They use state variables from steps 3, 6, 9 and 12 and accumulations from steps 12, 15, 18 and 21 (see Fig 1 of that paper). This inconsistency is because the fluxes (accumulations) are particularly prone to spin-up artefacts in the first few hours of the forecast. The trade-off is that the states and the fluxes used in the forcing come from slightly different forecasts.
