Is it possible to mine ZCash from a virtual machine with Linux?

Well, is it?

One other question: Is ZCash similar to Bitcoin Core where I'll have to download the entire block-chain?

u can easily mine it from bare metal, VMs and containers. entire blockchain is not needed to run miner.

Perfect. I think I'll just install it on a fresh HD on my other PC. It's more secure when you get a wallet that downloads the entire block-chain, correct? I'm somewhat of a newb to crypto-currencies, and a complete newb to mining. I'll need a wallet for mining, correct? Any suggestions?!

wallet and miner can be (usually are) different apps.
you can mine with zcoin app (which provides you with a wallet) but it can't join pools (solo-mining is something you want to avoid - I assume you don't have enough powers since you say ur noob).
To mine you need a wallet which you can get with zcash app - it's usually used to identify u on a pool and also (primarily) to send you mined (nano)coins.

On the ZCash website, though, it says it was designed in such a way to prevent centralization or farms and promote individual miners. So solo mining is out of the question? A pool as in a collective of people who pool their computing resources to mine coins, which then splits up the coins amongst the collective once they're mined? I do have a Bitcoin Core wallet, but I'm assuming that's only for Bitcoin, correct?