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How rigorously is conjunction analysis (finding out the chances of hitting an asteroid or another known object) conducted for missions beyond Earth orbit?
Due to larger computational resources available has there been (in recent years or decades) a corresponding increase in attention devoted to ensuring a deep space probe won't hit another body accidentally? I know the space is VAST, but so is human complacency.
Are there any fixed rules that are obeyed by mission planners to avoid conjuctions?
For deep space missions, there usually is a conjugation analysis, but not for the reasons you predicted. In fact, they do it to see if there's any secondary objects that can be studied while on the way to their destination. The majority of asteroids that have been photographed up close, in fact, were such accidents. From the Wikipedia Article, here's a few examples:
The first true asteroid to be photographed in close-up was 951 Gaspra in 1991, followed in 1993 by 243 Ida and its moon Dactyl, all of which were imaged by the Galileo probe en route to Jupiter.
The chances of impact, however, are so tiny that it isn't really worth discussing. Think of it like this. Earth gets hit by a rock large enough for us to see is in the thousands of years. The odds of them hitting a spacecraft are even lower.
