Is the "Distributions for eligible emergency expenses" limited to $1,000 per calendar year? If so, the system allows any dollar amount to be added.

Yes. If you weren't in a federally declared disaster area, and you took an early distribution from your retirement account for an emergency, you are limited to an exception from penalties only on a withdrawal of $1,000 or less.  You should only enter up to $1,000. 

If you were in a federally declared disaster area, when entering the income from your 1099-R, check the box for "I took out this money due to a qualified disaster" on the page entitled Do any of these situations apply to you? and continue to enter your information.

To read more about early distributions, see this article: Are there exceptions to the early distribution penalty on an IRA?