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Yes, you can [do addition with a slide rule]. Use the C, D, and L scales.

Sorry, no. Absolutely no. (Unless you mean adding logrithmic values, which is what the C/D scales are all about, but that's for doing multiplication, not addition.) Simple addition could be done, however, with a pair of L scales, or any other 2 scales with equi-distant digits. Then it's just like measuring a distance by using multiple lengths of a ruler.

You can do an addition with a regularly notched stick. Say you want to add 259 to 2,487. You just count 259 notches from one end, count 2,487 notches from that point, then go back to the end and recount how many notches total it is from the end to the place I forgot to tell you to mark. Of course, it takes a really long stick to do big numbers so I like to use a knotted string.

I carry such a string at all times. People are always commenting on the bulge in my trousers.

Sorry, no. Absolutely no.

O.K. Olympic one. But I just did it.

As you note the Slide rule is actually an adder. It adds lengths. The lengths happen to be measured on a logarithmic scale. The L scale allows you to generate a linear scale. As Vade noted in 1517 with his notched stick(that must hurt) addition is trivial when the stick is so marked. The L scale is linear. Find the first number on the L scale and put the index(left or right) of the slide(C scale) over that number. Next find the second number on the L scale go up to the D scale and read the value. Next using the just found value go to that value on the C scale. Drop down to the L scale and the answer will be there. QED.

There may be some gymnastics involved with the hairline, index, and a left carry. But for 2 number addition you can only carry a one.