Does using various app stores in parallel cause any issues? 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?
Depends on the store you're using, and how safe it is. See e.g. our alternative-markets tag-wiki, questions using that tag, and especially How safe is it to use Aptoide?

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).