My 6 an7 are the smaller, faster cogs...surely not worn already as only 18 months old as is chain.
Sounds to me like worn chain and cassette, its not so much how long the chain has been on the bike, but how much mileage you have done and to what standard has it been maintained. I used to wear a chain out on average every month and half, aprox 1200miles, but since I changed my lube I now get nearly 3 months and roughly 2400 miles.
But as a rule, I have always found in the past, that if you just run a bike on the same chain, chain rings and cassette, I do not have a problem with skipping. My Raleigh road bike was over 20 years old, worked perfectly till I scrapped it, the chain was well worn out, but it never skipped. My 10 year old MTB, has never had anything changed, yet works perfectly, I did change the chain and it skipped like hell, so I put the old one back on, and as it is only for commuting to work in the snow and ice, I'll just run it to destruction.
But one of my newer road bikes, only lasted about 6 months before I had to change every thing as all the transmission had been worn out and skipped like crazy. Maybe they don't make them like they used to do.