These are all really valid points!
Practice - For which ever role you play, be confident that you can be successful with several champions. Know who counters these champions, and be familiar with their kits. I improved my rank play by a ton when I started playing ARAM, and learning how more champions worked.
**Downplay the stakes ** - Sure, losing sucks, and everyone wants to win. Sure, it's going to be more work to get back into promos if you do lose, but so what?! You've done it before, so you're totally capable to do it again! I second the point of rank not mattering in a game ~ it's definitely all about how you and your team plays together.
View every game as a less, not as a test - YES. YES. YES. Okay, so you lost. What can you take away from that? Did you not have vision, and were too pushed too far out? Was it a bad Baron call? Did you try to 1v1 someone and realized just a little too late they were fed? What could you have done differently? What did their team do well? I know it can be hard to look into a loss and critique yourself, but it'll help you prevent the same mistakes from happening again, and will ultimately improve your play!
How I avoid getting nervous - For me, shaking off the nervousness was as easy (hah) as learning multiple lanes. I would have a panic attack when I first started queuing in ranked because I only knew how to mid. OH GOSH, WHAT HAPPENS IF SOMEONE CALLS MID, THEN WHAT?! SHOULD I JUST INSTALOCK?! Once I learned several lanes, it was a lot less stressful in champion select, and it makes it easier for people to fill, creating a happy team, not three people who can only mid, and now it's totally an ARAM!
What also helped me was just learned how each champion worked, what their skills were, what they were strong against, what they're weak against. The more you know, the more confident you become, the more confident you may feel to lead your team in (or out!) of a team fight!
If you're still nervous, try practicing in a draft pick, but treat it like ranked. Play how you normally would ~ take it seriously. It's a great way to improve your play without worrying about losing your rank!
Good luck!