How to export or backup Gmail messages?

I’m honestly quite conflicted about which topic to tackle here 😁
1) Is there any way to actually back up—or even just copy—emails from Gmail? I run my own business, and practically all of my professional correspondence lives in my inbox.
We’re talking roughly 1,000 emails at this point... and frankly, almost every single one matters to me. But I can't shake this nagging anxiety that something might go wrong. You never truly know when a server might fail or an account gets compromised.😛

2) On another note, I’ve sent out somewhere between 50 and 70 emails recently—mostly business proposals sent to various companies. Most of them were directed to general addresses like info@.......
Is it possible that my messages are just getting dumped straight into spam folders? It’s been over four weeks and I haven't received a single response.🕺 ?!?!
Most of these businesses are located elsewhere in the USA... different states and such...

Thanks in advance for any insight

There are a few different ways to handle that first issue:
http://www.labnol.org/internet/save-...le-docs/21045/
https://productforums.google.com/for...il/9V2yJ-H8gGg
http://lifehacker.com/5990556/how-ca...ersonal-backup

Basically, the solutions boil down to two main approaches—either saving your emails as individual files (like PDFs with the attachments included) or just connecting your Gmail to an external mail client—maybe Apple Mail or Gmail—and then exporting everything from there, or just using the client itself to manage the backups.

As for the second issue, I guess those emails probably ended up in spam because you sent a huge batch to so many different addresses in such a short window of time.