I 'inherited' a small Android tablet from a sibling, a Samsung Galaxy 4 SM T230. Old PC geek that I am, I haven't found much use for it yet, apart from reading PDFs. But I wonder if there are any good games I could play on it that are either exclusive to mobile devices or very suitable for them, despite being available for PC as well. I'm thinking something in the vein of Monumental Valley (exclusive and highly acclaimed) or Lara Croft Go (available for PC but made with mobile devices in mind). So my questions are:

1. What are some of the better games available for Android? Games that are interesting enough for a PC geek to pass some time on the train and that don't have necessarily superior PC versions? And that aren't too annoying regarding in-App-Purchases and Ads?

2. How can I determine which games will run fine on my tablet? What specs do I need to pay attention to?

3. Are there any good review websites where you can find information on how In-App-Purchases affect a specific game?

4. Is there anything else I could use the tablet for, apart from internet surfing and streaming? Any useful apps?

5. I heard Amazon abandoned their "free App of the Day" program - are there any other options to get free or cheap apps of good quality to try out?

They actually replaced it with Amazon Underground, which does have free apps without adds or money purchases (you should be able to buy any in-app purchase without actually getting charged, but I haven't had the need to try it yet).
There are quite a few titles in it, though you'll probably need to go through a lot before finding something you like.
P.S. You will need to install the relevant app on your tablet, and you may need to occasionally log in for your underground apps to work, which may or may not be a deal breaker for you.

I hate the amazon app store after they forced you to use one click.

You can set Amazon App Store to stop in app purchases so your kids don't make unauthorised in game purchases. However the app store has no such restrictions, they can just tap a game and buy it, no security, no PIN, no Password, just instant buy.

They only way to stop it is to use a third part AppLock program to lock the App Store. Insanely stupid.

As for games on the Android.

There are some good point and click adventures ported.

Personally, I use one of the CC numbers for testing and set that as the default payment, which gets automatically rejected. You may have to talk with Amazon support if it does get used, but you should be safe from charges.

I contacted Amazon when they first introduced this "Great" idea, said how stupid it is. The Support staff agreed.
I contacted them again a year later when the AppLock failed and kids spent about £30 on games (got full refunds) , then another year later.

If Amazon can't be bothered to enact a simple security measure, I can't be bothered to use their system.

I shortly got my hopes up, noticing that Amazon.com has Monument Valley and others as "actually free" with Amazon Underground, but it seems those offers are regionally restricted. German Amazon.de only has about 5 pages of run-of-the-mill games that look no better than the average Play Store shovelware ...

No suggestions for 1)?

I can strongly recommend Highrise Word Heroes - that's my favourite "on-the-bog" game right now. However, I got mine from a Humble Mobile Bundle, and it is balanced and designed to go without in-app purchases. It's also ad-free. As far as I can tell, if you get the game on Google Play, it's the same game but with ads and a lives/timer system, but you can pay an IAP and that will ensure that it remains free of ads henceforth (so it's not a "microtransaction" per se, but an "activation" fee that is persistent). I believe the premium version will set you back $3.

Beyond that, good mobile/tablet games are a veritable rarity. The Final Fantasy games work quite well on Android, but they're also on PC (arguably with minimally better graphics)

Thanks for the suggestion! I already played Highrise Heroes on the PC, but I agree, it was fun.

Another one I have on my Humble playlist is "The Adventures of Poco Eco". I've had it installed for a while on my phone and it seems designed to be played on a touchscreen, although I've not really been able to get my head around the game.

Otherwise...

Canabalt (also available on PC, but just as good on Android, especially if you have some kind of physical button on your device that can serve as a jump button - I find that the screen is just a tack too laggy for the reactions necessary)
The Room 1 & 2 (also available on PC with much shinier graphics, but it's designed first and foremost as a touchscreen game and it loses a lot in translation to a mouse-based control scheme).
Fieldrunners 1 & 2 (identical to the PC version, I find touchscreen works better - caution: the Humble version of FR1 has problems with recent versions of Android and the developer has been ignoring my contacts regarding this).
AaaaaAAaaaAAAaaAAAAaAAAAA!!! (Force = Mass x Acceleration) (the developer is the same shitsack that left Ugly Baby on Steam unfinished, but it's a fun game with a bizarre sense of humour, even if it's a little rough around the edges)
Osmos (identical to the PC version, I find touchscreen works better, especially given the fast reactions needed)
Funky Smugglers (there is a DRM-free PC version floating around out there, but given that multitouch support is needed to play the game properly, phone/tablet is the only way to go with this one)
Tomb Raider 1 (only recommended if you have a functioning gamepad - reason being that this version is so much better than the DOS version)
Aralon: Sword and Shadow (an open-world RPG that has the distinction of being one of the few "big" games to have never been ported to anything but iOS and Android)

First, here's where I found most of these apps. It matches mostly to the F-Droid repos.

So I'm going to be honest, this is an endless runner mad as an injoke for a certain Twitch Streamer.

Here's a simple solitaire collection.

Remember those crap brick games? Enjoy them on your phone!

I am Level has the style of a ZX spectrum game and is a bit of an odd mix of pinball and platforming. Tilt!

It's an open source chess engine, hurrah.

Sid the Snake is a very interesting and kinda twitchy twist on Snake.

Simon Tatham's Puzzles is a port of puzzle games from old palmtop devices. 39 puzzle games in one nice, small, ad free package.

Vector Pinball is a very lo-fi Pinball game.

Robot Finds Kitten: A nerd's zen garden.

Retroarch: When all else fails, video games.

OpenTyrian. I shouldn't have to explain this.