Buy a Shield TV. It's THE device to get if you want to game.

As far as the monitor option, what are you going to do about sound?

Question: Does Android Smart TV support all the Android apps?

No.

The regular touchscreen/tablet/smartphone Android Play Store is different from the seperate Android TV OS Playstore. They are not the same.

Regular Touchscreen Apps are designed - well - for touchscreen devices. Not TV's.

Android TV OS Apps are designed to be used with a simple D Pad remote, left, right, up, down, OK and back buttons.

Android Touchscreen/tablet Apps Sideloaded into Android TV OS devices needs to be used with either a seperate purchase tablet remote control or a wireless air mouse remote, both of which are a right Royal PITA to use with a TV screen sitting a distance away on the couch. Not recommended.

To make matters worse most of the tablet Sideloaded Apps display in landscape mode, so a seperate Rotation App also has to be Sideloaded. Even then these tablet Apps may not work properly. Sideloaded Apps do not auto update either, so there is ongoing maintenance.

In short you asking for potential trouble and ongoing frustration, it really depends on the Sideloaded App.

It depends. If a box runs 32-bit Android TV, you can only install apps with 32-bit support. 64-bit only apps, like Dolphin emulator, requires devices with 64-bit Android TV. 64-bit APK can't be sideloaded on 32-bit Android. At this point, the majority of apps on Play store still supports 32-bit.

Apps also have requirement on minimum API level ( https://en.wikipedia.org/wiki/Android_version_history ). An APK won't install if device OS doesn't meet targeted minimum API level. Since most Android TV boxes on the market now are at least on Oreo, this shouldn't be a problem for a while.