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It depends on the industry. Some are very desk heavy fields, others are very field/lab heavy. The majority will be desk jobs. There are relatively few jobs for full time lab engineers. The majority of the work in an engineering project will be done on a computer.

Eventually it will need to be taken to a lab to be tested. There are certainly positions like test engineers, that specialize in testing methods in a certain field. However, you're not doing much design in that position, you are simply taking other people's designs and testing them according to some standard. Test engineering has its benefits: its more hands on, but it's less intellectually stimulating and you don't get to do any design work. However, it's a bit more fast paced and you get to play with some toys.

Again, it varies by field. Environmental, petroleum, civil... some of these positions require a lot of on-site work (and travel, for better or worse). However, there are certainly no guarantees, plenty of civil engineers have desk jobs. Others, such as AE, ME, EE, will be more desk jobs. There are always exceptions, ME and ChemE often work in factories and aren't necessarily desk jobs.

It depends on what you want to do. If you want to work in a factory, you can. If you want to sit behind a desk, you can do that too. But if you wanted a simple answer: the majority of engineering jobs are desk jobs.