While the last part is true (about having to find a new employer quickly if you are fired or laid off), it's not true that you are stuck with your sponsoring employer. H-1Bs are allowed to switch companies. The caveat is that if the previous employer had started the Green Card process, it is reset with the new employer, unless the employee is at the last stage of the process (adjustment of status) and has been in that stage for at least 180 days.
> Even if you do keep your job it is extremely difficult to convert it to residence (I believe that requires costly employee sponsorship too).
All of the big tech companies apply for Green Cards for their foreign employees. Some have a requirement that you work for them for some amount of time (I've never heard of more than 1 year), others will start the process as soon as you join.
However, there's one BIG gotcha for someone coming from India: they will have to wait for a really long time to get their Green Cards. I'm talking about 10+ years here. For China it's bad too but no so bad. For the rest of the world it used to be 3-5 years (for EB-3, EB-2s have had it quick for years), but recently they sped it up a lot. I'm from Brazil and it took me a little less than 2.5 years to get my Green Card, counting from the day I set foot in the US. The actual process was a little more than a year.