afaik, かけずに just means to do something without "かける" (depends on what it is - like メガネを掛ける, that becomes メガネを掛けずに（出かけた）or 金をかける（I always thought かかる was more common though) becomes 金をかけずにすむ)

I guess it can be roughly translated as "without". Any native speakers to correct me on this?

And I think said native speakers can probably explain the second one better, I don't quite get it with this amount of context. All that I know is 浮く can also mean "stand out" or "stick out" or "be conspicuous", usually with a slightly negative connotation I think (related to the image of "floating" out). For example 主役の木村拓哉だけが浮いている means something like "Only the lead role's Takuya Kimura really stands out".