There are some Wikipedia bots that I consider to be absolutely indispensable: they perform tasks like fixing broken redirects, automatically reverting vandalism, etc. Is it possible to use these Wikipedia bots on other wikis? On large wikis, manually reverting vandalism and fixing broken redirects is a rather tedious task, and ought to be completely automated.
That always depends on the bot in question, there is no simple way to take several completely different bots and make them run on your wiki.
Some of them already operate on multiple wikis, so if you contact the bot operator, he might just add your wiki to the list. Even if the bot doesn't currently work on multiple wikis, he might be willing to modify it for that.
Most of the bots are also opensource, so you should be able to get heir code, modify it for your wiki (probably just change the URL in the config somewhere) and run it yourself.
Most of those bots use pywikipediabot framework , and in those cases, at least for some of them, the only thing needed is to configure it to edit on non-wikimedia wikis, simply by creating a configuration file.  Other frameworks have different methods, but at least the tasks mentioned are tipical of pywikipediabot, fixing broken redirects for example, is a standart script that comes with pywikipediabot. 
I can't add more than 3 links, so  and  are the
If you think a bot operation is of general usefulness for a MediaWiki wiki, not only you can reuse the respective code, but you could and should also try to get it included in MediaWiki proper.
The example you mention, fixing double redirects, is an example of an operation that now MediaWiki itself can perform, if you set
$wgFixDoubleRedirects to true (see warnings).