Could someone please help me with defining what it means to be a sound designer/ mixing artist working on a feature film as well as what the role is of the sound designer, his/her responsibilities, the working environment and employment conditions, entry level conditions with the potential of growth and the pros and cons of working in this field.
Second, this is an exceedingly broad question (or series of related questions tied together).
Sound designer can mean one of two roles: the more or less 'incorrect' usage of the term is an individual either doing all or multiple sound editorial roles on a show (e.g. if you're the only one doing post on the film, you may go credited as Sound Designer instead of Supervising Sound Editor, etc)
The more accurate definition is that they are someone who is charged with designing and creating never-before-heard (usually) concrete sounds (and/or an expressive library of these sounds), as well as creating conceptual and subjective (think whooshes, stings, transitions and all those) signature sound textures for a show (or a series, if we're talking episodic TV).
For the purpose of answering your question, I'll go with the latter definition.
Their responsibility is to ultimately develop the sound signature of a show, creating all of the non-Hard FX effects elements (and sometimes Hard FX-related elements) and in many cases nowadays, also required to edit those together for the show in question. Sometimes this involves creating specialized, literal computer sounds for a high-tech system (think Minority Report), other times it's creature development (think Super 8), sometimes it's sweetening combat with larger-than-life, in-your-face ballsy sound textures. And other times it's creating scare moments, stings, transitions, whooshes, swishes, booms, speed ramps, flashes, whip pans, snap zooms, suck outs, swells, drones, bumps and other tasty elements to work cohesively to drive/reinforce the story and visual where appropriate and where requested by the client.
As far as entry-level goes, my best understanding and from my own experience, is that it requires some background experience in cutting sound effects and working your way up. It's not one of the walk-in entry level roles (in the feature world at least, as per your question). It's a considerably demanding role and comes with a large responsibility. Sound Designer sits in a gray area between Sound Effects editors and the Supervising Sound Editor - as what they are doing is sound effects related, but they are charged with making sweeping creative and aesthetic decisions in tandem with the supervisor. It's a pay your dues type of role that some individuals grow into by chance or by choice. But usually you have to show that you can deliver solid tracks in regular sound effects editorial for some time and the like in order to be provided more opportunities to instill creative freedoms and subsequently ease into a role of a Sound Designer (that's been my own experience).
As for pros and cons, I can't really answer that because the scope and esteem of a Sound Designers role changes on a show by show basis and it's one of those "to each their own" opinions - which too change on a show by show basis.
The working environment also depends too. I work from a home studio, as do a lot of other sound editors I work along with and/or know personally. That seems to be the paradigm today. On major Union shows, depending upon how classified/big the show is, you might be working on a studio lot in an edit/design suite. Again, it all depends, so there's no definitive answer to this question. Employment environment and conditions are transient in nature.
Hopefully that helps! Again, only my opinion.