Lots of people use a VPN (like OpenVPN; PiVPN uses OpenVPN and is really easy to set up), but it doesn't get you away from needing to have a computer of some sort on all the time to act as the server--but that can be something as small and unobtrusive as a Raspberry Pi. Your router might also have a VPN built in, so I'd check out that, too. What you need is a VPN server, not what a lot of people think of when they hear "VPN" in the sense of you being a client to mask your location and whatnot. This is more like a workplace VPN that would give you a secure connection to your home network when away. But TeamViewier might be easier aside from the need to have a full-blown computer on.
Assuming you mean the built-in Button Controller app, that's probably what I'd recommend. If you need more power, Rule Machine with a "Button Device" trigger type (not "Button," unless you have discovered that this is what you want) will give you a similar UI with Rule Machine features like variables. But I wouldn't start there. Is Button Controller lacking something you're looking for?
There are also community apps that may have a GUI you like better. Advanced Button Controller is frequently recommended. However, Hubitat's own Button Controller app should be able to anything ABC can do nowadays too.
Up to you! Personally, I like to use my devices' "native" integration with Alexa (I don't have GH, but same idea) whenever I can. That way, I'm not putting Hubitat in the middle for no reason, and it's a tad more reliable (not that Hubitat isn't, but if I'm applying an update and rebooting my hub at that moment, I can still control my Hue lights, for example). Sometimes this also gets you more features, though Hubitat has been expanding the types of devices allowed via voice integrations a lot recently. You can choose exactly what Hubitat devices to expose to Google Home or Alexa, so you'd just want to be careful to avoid choosing duplicates.
But there's no right or wrong way to do this.
A few ideas. First, look into the tile size options. By default, I think their width and height is specified, but you'll see that if you set them to 0, they will auto-size. On the flip side, this means you'll need to pick a number of rows and columns appropriate to the size of your screen. (Or you can just set width to auto-size, set a specific size for height, and just scroll up/down when you need to.) While you have three devices, maybe their screen sizes are similar enough that at least automatic width (and maybe height too if you can) will be enough to help.
Second, there are some community Dashboard "enhancers," Smartly being probably the most popular. Some (all?) people behind it got kicked off this forum for unrelated reasons, so you'll find more current information on that elsewhere that I'm afraid to link to, but some searching should help. 
But if you're not unhappy with Dashboard after making the above modifications yourself, you don't really need this.
Sounds like you're on Android, and I don't know exactly what options Android users have found there. (On iOS, you could set up HomeBridge and integrate Hubitat devices to that, then use the various methods Apple allows for controlling these devices on mobile.) But one option anyone can do is create a Hubitat Dashboard with specific devices and then "pin" a link to that Dashboard directly on their main/home screen, no app needed (you don't need the Hubitat mobile app to access Dashboards; it's just a convenient way to get to them--they're just webpages, and the Hubitat Dashboard "child app" for that specific Dashboard in the Hubitat admin UI will give you a local or cloud URL you can use to directly access it instead). But I'm guessing there are more creative solutions others have found.
Also, as is tradition to say, "Welcome to Hubitat!" 
