Nobody sponsors citizenship. They can sponsor you for a green card (permanent residency). It costs thousands of dollars (I have been told in the range of $12,000, without considering attorney's fees). But it locks you in with the employer for several years, and when you consider the amount of money that you can generate for a company, combined with the fact that you will probably get an average or below average salary to compensate for the green card costs, this is not an absolutely terrible deal for the employer. You have to have some valuable skills that justify the employer knowing right away that he/she wants to keep you for several years!
I worked for my employer for about 6 months when we started green-card proceedings. I was a junior engineer on an H1-B then, having been hired straight out of college from the Netherlands. The job-req I applied for had been sitting there for well over a year, no American worker could be found to fill that position (many interviews were done, no suitable candidate found). I got my green card (EB3) in January of this year, 5 years later and am now the lead security architect on a large project. In those 5 years I've never been paid below average. I'd say it panned out quite nicely for both me and the employer.
That's a good question. I had a hard time getting letters from my previous employers, so that part took the better part of a year. If you ignore that, I'd say 3 years? I was EB3 'other'. EB2 or EB1 category applicants from 'other' probably end up taking about a year. If you're unlucky enough to be in the India or China region, you're going to be waiting much, much longer than that though. EB3 India is probably more along the lines of 7-12 years.
Edit:
I think it's sort of important to note that I came to the US in 2010, which was a weird year for H1-Bs. The economy was bad, and I was able to file for an H1-B in August after I graduated. This lack of H1-Bs in 2010 probably resulted in fewer GC applications down the line as well, so your mileage may vary.
EB2 'other' anecdote: I came to the US in 2009 on a H1B (job offer in October, granted in December) and filed for my green card almost immediately. I had my card within 18 months, most of which was PERM.
I realise I was very fortunate in three things: timing, nationality (British) and education (two Master's degrees, so EB2). During this time I was paid market rates, though I did go through a period of minor panic when I realised I basically couldn't leave Google while my green card application was pending and I had no idea how long it would actually take. But articles like this make me extremely sad that others can't be so fortunate, and I wonder what it is we (non-voting residents) can actually do about it...
These days generally labor certification in STEM jobs takes about a year. PERM sped things up dramatically and now that immigration lawyers are comfortable with how it works it generally takes less time than the old system.
Adjustment of status is where a lot of people get stuck. Really this should take about a year, but if you come from a country with a lot of migration (Mexico, India, and China) you could be in for a lot wait, especially in EB3.
At best you are looking at 2 years, at worst I've heard upwards of 10 years. A good immigration lawyer should be able to give you a good estimate based on their experience and current USCIS workloads.
