Does downloading and installing (although not necessarily running them all at the same time) a lot of apps from Play store slow down your phone?
The OP asked this question 3 months ago, and never came back to comment on the answers, but let's look closely at what he said, and the possible situations:
1. He says "does downloading and installing" slow down your phone. Although one previous poster answered that yes, the phone slows down a lot while downloading apps, I don't think that's what the OP was really asking about.
2. He says "not necessarily running them at the same time." This leads me to think he's asking if simply having a lot of apps installed on your phone slows it down. And the answer to that is absolutely, completely, "No." Apps are installed to ROM storage. Running apps run in RAM. Those are completely different, separate things, and one has no effect on the other. You could fill up 15.5 Gig of your 16 GB of ROM storage and it won't slow what running in RAM at all. The only catch might be if you run an app that attempts to write data to storage, and there's no free ROM to write to. In that case you'd get an error of some sort.
3. Does having lots of Apps in RAM slow things down? It depends. If you get a tool that lets you look at what's in memory (I highly recommend System Panel) you'll see that apps that are in RAM can be Active or Inactive cached) apps. Inactive apps are in memory so that if you want to access them, they are available immediately. They use no CPU, no battery, and don't slow things down at all, except in one situation: If you start a new app, one that isn't in memory, Android will flush existing app(s) out of memory to make space for the new one. There is a measurable, but hardly noticeable, delay in loading the new app while that happens.
Active apps may or may not be using CPU at any given moment. If you've recently used an app, it will show as active, even though you're not using it at the moment. Eventually, if you continue to not use it, it'll move into the inactive (cached) category. Some active apps may be running in the background, performing actually processing activities (such as downloading a web page while you're looking at another app). That does use CPU, and will slow other things down. Most apps don't really do much of this, however. Some active apps are just sitting there, doing nothing but waiting for a request from the OS to do something. Their cpu usage is minimal, and any impact on performance also minimal.
Unless you have an app that's doing lots of background processing, which is pretty rare, I think, having a lot of apps running doesn't have a serious impact on performance either.