Yes, you can open any number of traditional / Roth IRAs with any combination of institutions that you want.
Yes, your new company's IRA will affect your other IRAs. The maximum that you can contribute to all of your IRAs combined is $5,500 per year (currently), so any contribution by you or your company to your "company IRA" will count towards that maximum. That's a huge step down from a real 401k, I hope you took the implicit pay cut into consideration when negotiating salary.
Why would a company offer an IRA rather than a 401k? Wouldn't they get more bang for their matching bucks if they offered a 401k instead?
Well I don't really know, but since you're asking me I'll hazard a guess...
I don't think it really make a difference to the company one way or another. A dollar that they contribute to a 401k should cost them just as much as a dollar they contribute to your IRA. It makes a difference to the employee, but not the employer.
Additionally, IRAs are owned by and managed by the individual, 401ks are managed by the company (to an extent). There have in fact been lawsuits against companies that have not been "living up" to their "fiduciary responsibility" of providing "good" options in their 401k plans. This is the reason that the vast majority of companies don't just have a whatever-you-want-free-for-all in their 401k plans, they are required by law to try to pick "good" options for you so that you, the poor stupid incompetent employee, don't bet your life's savings on penny stocks and loose everything.
So you add the company's indifference on the cost of the benefit and the extra hassle, responsibility, and liability that comes with offering a 401k, and I'm not terribly surprised that there are some companies (generally small ones I would think) that would decide it's not worth the effort and end up screwing their employees over a little. And really, if the choice for them is either match IRA contributions or have nothing at all, then I guess it's better that they do offer to match your IRA contributions.