Regular IRA and a new 401k

Off the top of my head, no. I am sure that a prior IRA can continue as is, but don't think you can add to the IRA in the same year you're contributing to the 401K.

If you like the 401K, you should be able to transfer the balance of the old IRA into it without affecting the 401K annual contribution limit.

Again, I haven't researched this.

I think you can as long as your total contributions to both funds doesn't exceed your limit for the year.