When another player builds next to you, you can choose to taken power (and lose the #power-1 points). But when the number of power you can take is greater than the number of tokens you can move to your third power bowl, can you immediately convert power into something else? Or do you lose the rest?
You lose the rest. Conversions are performed only on your turn. According to the rulebook (p. 13):
Anytime during your turn, on top of your Action, you may do any number of Conversions.
and from p. 9:
Once all Power tokens are in Bowl III, you cannot gain further Power.
Therefore, if you are close to the limit of your Power, and expect to gain more before your next turn, you may want to consider Converting some of it to a useful resource.
You indeed lose the rest. But keep in mind that you only have to pay VPs according to the power you actually get, so in contrast to what @sitnaltax suggested I'd rather advice that you do not covert power into money or workers in order to be able to gain more, unless you are in desperate need of that resource.
1 free power is a no-brainer
2 power for 1 VP is usually a good choice until round 4 or 5
3 power for 2 VPs is usually ok during the first 2 or 3 rounds. In the second half of the game it is rarely taken - only if it makes the difference between being able or not to take a particular power action in the very next move.
4 power for 3 VPs is something that is hardly ever taken (less than once per game)
I have rather seen players deliberately not spending power in order to only have the capacity to gain 1 or 2 when their buildings next to the presumed building site of an opposing player would have entitled them to gain 5.
You said "you only have to pay VPs according to the power you actually get" but the example in the rules says "The Mermaid player (blue) needs to inform the Nomad player (yellow) that he may gain exactly 3 Power due to the newly built blue Dwelling..." Which seems to indicate that it is all or nothing in both power and cost. Nov 1, 2018 at 16:49
@EricAnderson: Yes and no - if you have enough power tokens in bowl 1 and/or 2 it is indeed all or nothing, you can not voluntarily take less power; that is what the cited rule says. But if you have almost all tokens in bowl 3 and can gain only an amount of power less than what you would be entitled to through the level of your buildings, then the "all" equals that maximum of what you can gain, and the VPs you have to pay are calculated according to that. It is still an "all or nothing" decision with that new, lower "all". Nov 1, 2018 at 16:59
I see the exception now: "If you have not enough Power tokens in Bowls I and II, gain as many Power as needed to move all of them to Bowl III and lose Victory points accordingly." Also that you can gain less power to prevent a negative score, although I've never seen that come up. Nov 1, 2018 at 17:07