This is a really interesting question, and it is one that is outside my expertise, but as it is ten hours in I will give it a god.

I cannot personally think of much in the way of direct competition. The exclusivity and intolerance of Christian sects is something of an innovation, and one that I personally think is a direct contributor to their success. There would be nothing odd is, say participating in the traditional polis religion while still being a devotee of Isis and maybe visiting the Eleusinean mysteries once in your life. Classical religion, an the Roman Empire, is sometimes praised for its tolerance, but I think this misses the mark: religious intolerance is the product of a very specific religious worldview, and praising something like classical religion or Chinese Buddhism for its "tolerance" is fundamentally an imposition and misunderstanding of the nature of religion. Roman conservatives may have occasionally looked down on "foreign" cults, but it wasn't because people were worshiping the wrong god, it is because of the host of associations carried by that god.

In practice, this means that it is very possible to have a very temporary participation with a particularly cult. The Bona Dea cult, for example, is well known for a scene in Apuleius' Golden Ass in which they whip and scourge themselves until the onlookers throw them enough money. These people watching may have no particular investment in the Bona Dea, but they could still participate in this particular ritual activity. The boundaries were fluid, if not nonexistant.

Another issue is that the different religious sects tended to target different demographics. The Isis cult, for example, tended to target the elite and well-to-do, although it was certainly more widespread than that. Mithras stayed quite consistently within the Imperial institutions, most notably the army and the civil bureaucracy: its fame today is largely due to those groups' vigorous epigraphic habit. The hyper-elite turned Neoplatonism into a quasi-religion lead by Iamblichus and his ilk.

To sum up, I suppose, the idea of religious "competition" is not something we can apply in an uncomplicated fashion to the classical world, at least not when in comparison to the modern one.

Allow me to come out of lurking to try and give you an answer as an historian specialising in Ancient Roman social and cultural history.

Probably the most important thing to keep in mind when you consider the Roman religious world is that religion and religious institutions operated in completely different manner within Roman society than they do in the modern world. There was no such thing as a formal hierarchical institution which might compare to the later Christian church, at least not centered around a single god in the traditional Roman system. The rise of foreign cults such as those of Mithras, Jesus or Isis is a different, but I'll get to that later.

This is one of the main reasons why there was no competition for political power between temples as you ask in your first question. Temples to different deities did of course have a number of priests, but there was no overarching society for priests to Minerva (for example). These priests could be very influential and enjoy a great amount of public standing. It did not however translate directly into policital power or influence. Roman politics was all about knowing the right people, doing them favours and receiving favours in return, with an eloberate informal hierarchy. So priests and supporters of a particular temple which had a high enough public standing (either through birth or a history of public office, usually both) might very well have tried to influence senators and political factions by 'befriending' them and pushing/asking them to support their cause. In this however it did not deviate from the way all political business was done.

Rivalery between temples or gods of the traditional Roman pantheon would mostly have revolved around very material subjects, such as funding for the renovation of a temple or the manifacture of a new cult statue. There was no need to jockey for influence over the populace since they played a very small role, if any, in the decision making process. It also mattered relatively little how popular a specific god was among the general population. Public gods and temples simply coexisted and the Roman people simply worshiped them all at once. Even when individuals might have had a specific preference, it mattered little on the grand scale. Again, religion worked very differently than today. A Roman might keep a small shrine to certain god in his house, but that had no negative impact on other temples. 'Winning souls' for a specific deity wasn't necessary and didn't happen.

This does not mean however that there we absolutely no religious instutions apart from small groups of priests of a specific temple. There existed so-calles Colleges of priests, such as the College of Augurs and College of Pontiffs. These weren't connected to a specific deity though. The College of Pontiffs were charged with overseeing a great deal of religious decisions. However, this did not make them a religious political force. These colleges were staffed by politicians, senators or aspiring senators, who saw these jobs as what they essentially were: just another set of public offices which could be used to gain public standing. There was no disctinction between religious life and public life when it comes to Roman politics.

This also explains how the Romans dealt with the emergence of cults like the ones you mentioned. It wasn't the temples that reacted; it was the state (either the Senate or the Emperor). How they reacted varied from time to time and from cult to cult. As has been pointed out, there was both acceptance and occasional rejection of new gods and cults. For example, the cult of Cybele/Magna Mater met with almost no resistance and was quite easily admitted into the Roman religious spectrum and became quite similar to the way the 'traditional' cults like those of Jupiter and Juno operated. Christianity, as we all know, had a bit more trouble.

This has everything to do with the implicit level of social status that was assigned to membership or participation in such a cult. At certain times is was considered extremely fashionable to worship Isis for example and the cult flourished in Rome and the whole Empire. Christianity only reached that point well after Constantine. It was the political/societal elite (from the first century AD onwards spearheaded by the Emperor) which created this social status.

As for you last question: it all depends on what you might consider the 'way the state was run.' Fundamentally very little changed to the system of emperorship, regardless of the religious whims of a particular ruler. There are however noteworthy cases of quite far-reaching fanaticism. Elagabalus has already been mentioned. Another example (although admittedly beyond the timeframe you mention) is the rule of Julian the Apostate in the fourth century, who attempted to crush the Christian faith and repopularize the pagan pantheon.

If you are interested in the general way religion was a part of the Roman world I can recommend the standard work Religions of Rome by Mary Beard, John North and Simon Price or Blackwell's A Companion to Roman Religion. A World Full of Gods by Keith Hopkins is also a fascinating read when it comes to the way religion and public life were intertwined in the Roman world.

In short: Cults surrounding specific gods held very little actual political power and had no reason to compete directly. The religious and public life were intertwined. Therefore it was the societal elite who responded to new cults / sects, not specific religious institutions.

In answer to your last question Elagabalus' devotion to Sol Invictus above everyone else was probably the reason why he was assassinated. He went and re-dedicated all the temples in Rome to Sol and really upset the Romans.