will my existing customers be able to install the new version with their previous license keys?
Yes. Your customers will only need new keys when you create a separate "major version" in LimeLM. So right now you probably have a "Your App" product with a single version "1.0". Those product keys you've generated with YourApp 1.0 in LimeLM will continue to work with your app for as long as you use the TurboActivate.dat and VersionGUID from within your app.
Some day you'll likely want to release a new "major version" of your app. That is, a new version for which these old "1.0" keys don't work. To do that in LimeLM, you just click the "New version" link (under Your App in your dashboard), enter all the details for the version, then click "Create version". Then you download the new TurboActivate.dat for that version (let's call it "2.0") and you use the new Version GUID in your app.
Does that make sense?
Moreover, would these re-installations count towards my subscriptions activation limit or does that only count number of machines?
Re-installations or re-activations don't count against you. The "activations" we talk about on the signup page are really just "activation slots". Each product key you generate can have 1 or more activations slots. Each slot can be used by only a single computer. But that single computer can re-activate in that slots as many times as you need.
So a good way to think about the "Solo" plan (the 300 "activation slot" plan) is that you'll be able to sell 300 separate product keys with 1 activation slot each. In other words you'll be able to sell to 300 separate customers.