I am using a Android phone with Lollipop and found that apps can be installed from various app stores like Google Play, Amazon App Store or F-Droid. Are there any problems I can run into if apps are installed from different app stores?
I've made pretty good experiences with F-Droid, and also with Aptoide (sticking to their main repository named "Apps", which is curated manually and well maintained – see the link in previous paragraph). Close to no side-effects as far as I can tell. Only issue with F-Droid is a two-fold: they build the apps themselves from the sources, so they need to sign using their own keys; while that makes sure you really get what it says (the
.apk is sure to match the sources you can investigate), you cannot "cross-update" apps from them with other app stores (e.g. install from F-Droid and then update from Google/Aptoide doesn't work due to signature mismatch).
I cannot speak for other markets, as I don't use them. One specific thing to consider are payed apps, for at least two reasons: 1) that most likely demands the market's app to be running on your device (for license verification) – and 2) if that market "shuts down", you will either have to buy the apps again from another place or no longer receive updates (which e.g. happened with AndroidPIT's app-center on 2014-12-31).
Like Izzy said, the security of the app store is very important. In addition, make sure not to download the same app, or an app with the same package name on two different markets, it might create a conflict and produce a few errors, possibly a signature mismatch but it may vary by market. If the creators of the market are smart, they would prevent the error by not allowing the user to download the app again, if they already have an app under the same package-name. Amazon and 1Market both prevent this issue. I am not sure about the other markets.