世界中をあちこち回りながら仕事がしてみたいって英語でなんて言うの？

This phrase implies that you are focused on a goal of yours to travel and work. If however, you used "I would like to" instead of " I want" this would suggest that you wouldn't mind and wouldn't be as serious.

Here you want to be able to explain that you want to be able to both travel and work at the same time.

You can be specific by stating that you want to travel the world while traveling. For example: 'I'd love to work while traveling the world'.
In some cases you don't need to be specific as the other person will most likely interpret this as wanting to travel the world anyway.
So you can say the following:
I'd love to travel and work and the same time
I want to work remotely while traveling

Seeking means looking for or pursuing. A profession that involves traveling would be a job or career that includes the requirement to travel as part of the job.

"Requires traveling the world" is a way to impress that traveling the world is a mandatory part of the job. This would mean that traveling is a big part of the job.

The expression "affords the time," in this case, means to allow the time. If time is afforded to someone while he/she is employed, this typically means that the job pays well and includes the option to take extended time off for traveling.

This sentence could mean that you would like to work remotely to be able to travel whenever you like. It could also mean that you would like a job that allows you travel around the world such as a flight attendant or cruise ship member. I hope that this helps. :)