Yes, Dallas is a home rule city. "The city shall have and exercise all the powers conferred upon cities by what is known as the Home Rule Amendment to the Constitution of the State of Texas and the Enabling Act relative thereto" from the City Charter.

Ireland

Home rule

"over 5000"

This answer is way off. It appears to be the answer for "how many residents does a town need to be able to apply for home-rule status?", which is 5,000.

The answer for the number of cities with home-rule status is closer to 300. The Texas State Historical Association reported in May 1994, that only 19 of the 309 cities in Texas of more than 5,000 population had not adopted a home-rule charter. Although this number is dated, it is obvious that there can only be a few hundred home-rule cities in Texas, since the minimum required population is more than 5,000. See the related link for home-rule charters.

Home rule cities.