Putting aside for a moment that the messaging market is pretty flooded with many competitors and that your app offering would need some unique differentiator (or an amazing marketing/PR strategy)... your questions that can be answered in a variety of ways depending on details you have yet to share. But speaking broadly:

Mobile app development costs vary depending on a couple of things:

1. Functionality (how little, how much)
2. Integrations (OS and external)
3. Backend / server-side hosting components.
4. Depth of testing and iteration.

You also have to realize that building an app and maintaining an app are two different costs.

You need to find an app developer or partner and define the above before you will have the costs you are looking for. Generally speaking the initial development costs could range from 50-150k (or more) depending on the developer/partner and the above variables. Before going all out and build a full feature app, you should consider looking at the base functionality and build a MVP

Building an MVP could cost as little as finding a solid app developer and paying them for their time.

With an MVP, a good business plan, and some solid market data you would be better armed to start the hunt for funding.