I need to do a dilution of 1:7000 and I need to make up 20ml of solution would 3ul into 21ml be correct?

Answer 1

I think so. The ratios or fractions should be equal.

3 uL * 1ml/1000 uL = .003 ml

so you have 1/7000 = .003/21 they both equal .000142

Answer 2

It depends on the accuracy that you need.
In theory it is correct and also in practice if you can tolerate small deviations, for example if you are planning to do immunoblotting.
If your application requires high accuracy then you should first do a 1:70 dilution (using sufficiently big volumes and equipment to be acurate, e.g. a 10 or 20 ul pipet to put 4ul of reagent in 276 ul of solvent) and then use from that an aliquot to do a 1:100 dilution. Although increasing the number of steps increases the error in any experimental procedure, diluting a very concentrated solution in a single step will lead to much larger errors because of the pipetting/volume handling errors.

It also depends on your reagent. If it is too expensive or you have limited amounts (as is the case for antibodies) then you need to do it in one step.