Do I need to run out and purchase a new MacBook Pro to upgrade to Xcode 11?

So I will need Xcode 11 by April of 2020, will High Sierra 10.13.6 support it?

If I download Mojave, will it support Xcode 11?

If I download Mojave, with only 11GB left on my hard drive, will that be a problem?

Xcode 11 requires a Mac running macOS Mojave 10.14.4 or later.

https://developer.apple.com/documentation/xcode_release_notes/xcode_11_release_notes

The download from the Mac App Store is 7.6 GB.

with only 11 GB left on my hard drive

I would suggest having more free space before trying to download and unpack the app.

• App Store: Not enough disk space to install Xcode
• Can't update Xcode even with 19gb free space
• Xcode free space requirement

The Xcode 11 release notes on Apple Developer website says it requires at least macOS Mojave 10.14.4 to run. In my personal experience, I was unable to run it on macOS Mojave 10.14.4 and was asked to update to 10.14.6.

Coming to the Mac requirement, any Mac capable of running macOS Mojave 10.14.6 can run Xcode 11. Subsequent releases of Xcode will eventually drop macOS Mojave and you'll need a Mac capable of running macOS Catalina 10.15.

It would be good to make sure that your Mac is capable of running macOS Catalina. Here's a list of devices capable of running macOS Catalina (source):

Answering your specific questions:

So I will need Xcode 11 by April of 2020, will High Sierra 10.13.6 support it?

No. macOS High Sierra 10.13.x doesn't support running Xcode 11.

If I download Mojave, will it support Xcode 11?

macOS Mojave 10.14.4 through 10.14.6 support Xcode 11.0 through Xcode 11.3.1. You may be prompted to update to a recent version of macOS Mojave (up from 10.14.4) for some versions of Xcode in the said range.

If I download Mojave, with only 11GB left on my hard drive, will that be a problem?

You may likely run out of space while attempting to install Xcode. While no official guidelines are published by Apple regarding required space, make sure you have around 20GB of free space on your hard before you attempt to install and use Xcode (based on generally observation using Xcode).