Mumat39 - not stupid questions
I do rinse veg that is going to be cooked, but I wouldn't go so far as to check really thoroughly for grit unless it's come from my mum's garden (not because I think it's dirtier, but because it's not been washed during a packaging process which would remove a lot of dirt).
Stuff to be eaten raw I do wash thoroughly (lettuce etc, I have a salad spinner which dries it well).
In the case of the cabbage I would remove the outer leaves but wouldn't bother washing inner ones - it's tightly packed and grit won't have got in, and I think most chemicals would be on the outside only. Plus it's going to be cooked through.
Spinach I do wash, as the water that clings to the leaves is enough to wilt it in. It's up to you how thoroughly you want to check stuff, but I haven't died yet and I definitely wouldn't bother checking every leaf - that does seem rather a faff.
I would never wash meat, as I understand that this can spray raw meat juices over the kitchen
On your kitchen equipment question, there is just dp and me, and we don't have a dishwasher. I would suggest getting yourself a decent frying pan (Teflon or similar), and a couple of roasting pans at the minimum (for roasted veg or meat) as well as a Pyrex dish (that you would cook something like shepherd's pie in) and maybe two or three saucepans of different sizes. This would set you up to cook most recipes that I use at least.
On the washing up front, dp does still eat carbs, so there is possibly a little more washing up than there would be if he weren't. In addition to crockery, there might be a meat pan and a veg pan/saucepan, plus a rice or potato pan/saucepan (also maybe a colander). However, I don't cook every day as I will often cook a large amount of meat and have it cold with salad for a couple of day afterwards, so the washing up for subsequent days is minimal.
The dishes that tend to create more washinnup are things like moussaka, where there would be a pan for he roasted aubergines, another for tehe meat mixture, another for the egg/ricotta topping, and then the baking dish where it's all assembled. But again, that would lay more than one night so I wouldn't have as much to wash up the next day.
Sorry for the rather long and rambling answer but I hope that helps!