2) Yes. Washing = a quantity of clothes, bed linen, etc. that is to be washed or has just been washed.
3) Neither. If you haven't bought them yet, then:
"I'm going to buy (a pair of/some) new shoes today" or "I'm going to buy new shoes today."
In the past it would be:
"I bought (a pair of/some) new shoes today" or "I bought new shoes today."
4) "I don't want to wear pants today" is what people actually say. The latter is not wrong but less common because it is longer for no reason.
5) Yes, that would be the same thing, but veggie bed is very uncommon.
Veggie/vegetable patch/garden are the options you would probably want to use.
If you use bed, it looks like garden bed is an okay option.