How can I build new rental property listing website ?

I can tell you a lot about that, because I'm making a complex listing website for Israeli real estate owner.
If you need "just" to list property and rent it out, then you probably can stick to a dedicated Wordpress theme. Try searching "real estate" on Themeforest. There are some themes that do exactly what you need — they list sevices in different cities through different agents.
If you want your clients to be able to book an apartment using an availability calendar, then you probably should take a look at the Pinpoint booking system. It is available as a Wordpress plugin, so you can integrate it directly into your listings, accept payments, set up an easy email delivery system etc.
Expect a lot of custom development to refine your website, though.

0. Install Wordpress.
1. Wordpress has tons of options and plugins that can make your website as SEO friendly as it gets. Take a look at Yoast plugin for almost all of your SEO needs.
2. I used Villareal theme for that Israeli project. It already built with an agent network in mind.
3. Use a social media sharing plugin from Wordpress plugin directory, there are plenty of them. They usually work out of the box.
4. Villareal theme does exactly that. You can create different paid packages which must be purchased to post listings. Banners are not included though, you need to manage them with some other plugin, or make a plugin hook that allows publishing banners when you purchase a package. (this means several hours of PHP coding)
5. This is somewhat tricky. You'll need an availability calendar solution (think of Pinpoint plugin) and you'll need to make your function to take property availability into account.
I had to code that functionality manually.
Maybe you'll be able to find another theme that includes that functionality if you can't code.

Use Wordpress. Get someone to install a premium theme from Fiverr.com

I would start with a Wordpress site. They are cheap to start and you can just purchase a ready-made template for real estate listing. There are even real estate plugins like IDX that can hook into your listings.

To create a new rental propery listing website, be sure to check out these wordpress real estate themes . There you can find products that can be used for vacation rental, real estate listing, or apartment rental websites. You can customize any of theme without writing any code.

Hi there! We specialize in building real estate websites (you may find us at agentdrive.com) and recommend that you pay attention to a few things:

1) choose a content management system and a real estate theme (our fellow members already mentioned WordPress and Themeforest which both offer tons of great stuff)
2) make sure your website is mobile-friendly so as not to end up penalized by Google
3) make sure your theme has a nice blogging template. Blogging is so important for SEO so pay attention to things like layout, newsletter/opt-in forms, social sharing especially as you already have a big follower base
4) check if the theme offers a great property search + include a reason why people should search for rentals via your site as opposed to large property portals (e.g., mention you have a greater variety of local properties and serve clients on a granular, hyperlocal level)

All in all, these are the things important from a marketing and SEO standpoint; technical side requires programming skills to set everything up.

Hope this helps!