score:9
This is a standard procedure. For US citizens (and many others) to get Russian visa an "invitation" is required. The party responsible to get this invitation is called "sponsor", and it is obtained from Russian MVD (ministry of internal affairs). To apply for "invitation" a copy of your passport is required (this site is official site of Russian Ministry of Internal Affairs; passport copy is listed in #2.1). So this is ok.
Now to your other questions:
(1) I'm unsure what a copy of your passport would be good for. Probably the most sensitive part of it is your signature which could be used to apply for a loan under your name. But if someone in Russia is willing to put this scam on you, it won't matter - corruption in Russia is such that whether a Russian court decides you owe money or not depends on totally different things. Which brings us to (2)
(2) You should certainly be concerned, but there is no need to be paranoid. Just be careful, stay out of politics and remember that free speech in Russia (this includes academic free speech) is different. As in the old joke:
George Bush talks to Putin:
Bush: you know, Vladimir, the USA has freedom of speech. Anyone can walk to White House and scream "George Bush is a moron and he should be impeached!", and the government would do no harm to him!
Putin: well, we have freedom of speech in Russia too. Anyone can walk to Kremlin and scream "George Bush is a moron and he should be impeached!", and the government would do no harm to him either!
One thing to be aware of is a much higher possibility to be misinterpreted. If you plan to give any interviews, expect your words to be misquoted, used out of context, or misinterpreted. Depending on your personality and area of expertise, this may or may not be an issue. The only way to avoid this is not to give any interviews to Russian media. This however would probably the worst case scenario.