You could use a supercoiled plasmid as the substrate. After the reaction, run it on an agarose gel and stain with EtBr. Measure the reduction in the amount of the supercoiled plasmid. At low levels of endonuclease activity, you will see conversion of supercoiled plasmid to relaxed plasmid. With more activity, you will see the accumulation of linear plasmid, then degradation products of smaller and smaller sizes. You could set an activity unit as the amount of endonuclease required to reduce the amount of a certain amount of supercoiled plasmid by half in a certain amount of time at a specified temperature in a specified buffer.
Beware that the endonuclease will be toxic if it is produced in active form and there are recognition sites for it in the genome of the host, hence you will get no clones unless the cloning procedure is performed under conditions where expression of the endonuclease is well repressed (and even then...).
The method that Adam B Shapiro described should generally work fine, assuming your endonuclease is not a site specific one (in which case you need to be sure the template has the proper site). There are lots of other nuclease assays as well, depending upon the specificity of your enzyme and the final products if the reaction goes to completion.
I would also remind you that many E. coli strains produce the periplasmic EndA nuclease, so be sure you aren't co--purifying that enzyme, perhaps by using an EndA mutant strain.
It depends. In many cases, e.g. restriction endonucleases, what is done it to clone the endonuclease together with its matching methylase, so that the host DNA is not cleaved (REs and their methylases are often physically clustered in the genome of the original host, which makes this trick easier).
I do not know how people go about cloning other, less specific endonucleases which might not have a matching methylase. In principle, you could try cloning it in a T7 system, so that all the cloning work is performed in a strain devoid of T7 RNA polymerase -and even in this case, you have to take care to avoid run-off transcription from upstream promoters. Then when time comes to express the protein, you'd use a strain where T7 RNA pol is more tightly regulated than usual (e.g. using the Walker strains, or cotransforming with one of the pLys plasmids, or using the RiboTite system), taking care to include glucose in all media to ensure repression, etc.
Also, depending on the specific characteristics of the endonuclease, you may have other tricks at your disposal. For instance, if memory serves well, NEB provides or uses a plasmid to check for the status of their SHuffle strains, in which an endonuclease which requires the formation of disulfide bonds is expressed in the cytoplasm. That endonuclease is not toxic to the host unless transformed in a Shuffle strain.