As standalone buys go, the Odyssey is not a good ship. You need to remember, the Odyssey has been nicknamed "The Space Whale" for a reason. She's slow, cumbersome, and can't turn for crap. However she's quite durable, and VERY versatile. But you cannot buy just one Odyssey and expect to do well. The best combo in the game currently for a non-dedicated tank Odyssey is the Tactical Odyssey + Chevron sep (which requires the Operations Odyssey). That nullifies a lot of turn rate issues (+8 base turn or something like that), and also gives you a huge power boost to your weapons (always useful especially when using beams). But you sacrifice a large chunk of hull and shields, and power to aux and shields in the process.
Your best bet would be the get the 3 pack. However if you are unwilling to do that, I would recommend instead the Regent or *throws up quietly in a corner* the Excelsior. Both ships do good damage (at least as cruisers go) and can tank nearly as well as the Odyssey (actually not quite, but it's close enough).
However for your other questions:
Sensor Analysis only requires that you be firing on the target and have it constantly targeted. Break target and it resets.
The Tactical Odyssey's +10 power alone is not enough to negate the drain from 8 beams (not much is). In all honesty 8 beams actually loses out on DPS compared to a 7/1 or 6/2 setup.
It is said the best weapon is one that is never fired. I disagree. The best weapon is one you only have to fire... once.