So I let my 8 year old cousins play a game on my tablet for a bit and next thing i knew they had downloaded and installed three new games, some of which acted suspiciously, displaying notifications every second until another app was installed for example. Is there any way I can check if these apps on the market are legitimate, or if my system has been compromised? I use this device to check my primary email and access bank statements etc.
Though I know of no definite way, I'd suggest looking them up on AppBrain. Additionally to the "Permissions" tab we had in playstore before their "Win-8 relaunch", that site offers a tab "Permissions & concerns", where some things are pointed out.
It might also be worth taking a look at their search results: Spam apps are "filtered" at AppBrain, so they show up single-line instead of full-heights in the result list.
While all that might only give indicators (and no definite answers), it's one step in the desired direction at least. A second step would be searching the web for reports. Including the app names (with links) in your question might also give you the chance somebody here recognizes the one or other.