Slipped gear would mean you over revved the engine by say, selecting 2nd instead of 4th, meaning the road speed for the selected gear is too high and when you let the clutch up the engine revs well over the clock, i.e. past 9,000 rpm. Since you haven't said you've done this, we'll ignore it. Honda will just be checking what error codes were on the ECU. They don't usually have to send it away though, they can usually read the code in the workshop.
With regards to the oil, I'm assuming you have an EP3 (2001-2006 Civic Type R) and I'm pretty sure it says this in the manual but I might be wrong, but Honda state that using 1 litre of oil in 800 miles is acceptable. So it's not the time you've had it but how many miles you did. Some cars use some, some use none. I recently used a litre at a track day at Spa and my car never normally uses any, so again it can depend on how hard the car is driven and what grade of oil was in it can also affect the amount of oil it uses.
The K20 only normally holds 5 litres of oil and it is 1 litre from Max to Min on the dipstick, so in a month you could well have got it well below minimum. If the AA said there was no oil in it, he means it's below min on the dipstick.
There could of course be an oil leak that has accelerated the issue, but I'm assuming they are checking for that as well.
Sorry to say it, but you've only yourself to blame really. Just like you check the air in the tyres and screen wash, you need to check the oil regularly in any car. Mis-shifts and low oil are about the only things that kill these engines, otherwise they are bullet proof.
