The errors with npm upgrade: it should be sudo before that line, that’s why you have a bunch of permission errore, but the node update took care of the npm update too.

To fix the mirror install, replace MagicMirror/js/electron.js with a fresh copy from github for your version of MM to undo whatever the patch broke when it failed.

npm install -g npm

as i typed in this, it started to install and shows this:

npm ERR! Linux 4.9.59-v7+
npm ERR! argv "/usr/bin/node" "/usr/bin/npm" "install" "-g" "npm"
npm ERR! node v6.12.2
npm ERR! npm  v3.10.10
npm ERR! path /usr/lib/node_modules/npm/node_modules/abbrev
npm ERR! code EACCES
npm ERR! errno -13
npm ERR! syscall access

npm ERR! Error: EACCES: permission denied, access '/usr/lib/node_modules/npm/node_modules/abbrev'
npm ERR!     at Error (native)
npm ERR!  { Error: EACCES: permission denied, access '/usr/lib/node_modules/npm/node_modules/abbrev'
npm ERR!     at Error (native)
npm ERR!   errno: -13,
npm ERR!   code: 'EACCES',
npm ERR!   syscall: 'access',
npm ERR!   path: '/usr/lib/node_modules/npm/node_modules/abbrev' }
npm ERR! 
npm ERR! Please try running this command again as root/Administrator.

npm ERR! Please include the following file with any support request:
npm ERR!     /home/pi/npm-debug.log

so i did it as root and rebooted my pi.

now i installed your module and got some warnings AND some errors:

pi@MagicMirror:~/MagicMirror/modules/MMM-RTSPStream $ npm install
npm WARN registry Unexpected warning for https://registry.npmjs.org/: Miscellaneous Warning EINTEGRITY: sha1-QFUCsAfzGcP0cXXER0UnMA8qta0= integrity checksum failed when using sha1: wanted sha1-QFUCsAfzGcP0cXXER0UnMA8qta0= but got sha512-zr6QQnzLt3Ja0t0XI8gws2kn7zV2p0l/D3kreNvS6hFZhVU5g+uY/30l42jbgt0XGcNBEmBDGJR71J692V92tA==. (260 bytes)
npm WARN registry Using stale package data from https://registry.npmjs.org/ due to a request error during revalidation.

> MMM-RTSPStream@1.2.1 postinstall /home/pi/MagicMirror/modules/MMM-RTSPStream
> ./install_sd_patch.sh && sudo npm i -g http-server

Applying Graceful Shutdown patch to MagicMirror.
error: Anwendung des Patches fehlgeschlagen: js/electron.js:96
error: js/electron.js: Patch konnte nicht angewendet werden
npm ERR! code ELIFECYCLE
npm ERR! errno 1
npm ERR! MMM-RTSPStream@1.2.1 postinstall: `./install_sd_patch.sh && sudo npm i -g http-server`
npm ERR! Exit status 1
npm ERR! 
npm ERR! Failed at the MMM-RTSPStream@1.2.1 postinstall script.
npm ERR! This is probably not a problem with npm. There is likely additional logging output above.

npm ERR! A complete log of this run can be found in:
npm ERR!     /home/pi/.npm/_logs/2017-12-22T17_39_55_666Z-debug.log
pi@MagicMirror:~/MagicMirror/modules/MMM-RTSPStream $ 

edit:

i took a look in the debug_log file after the reboot of the pi (2017-12-22T17_36_34_105Z-debug.log) :

0 info it worked if it ends with ok
1 verbose cli [ '/usr/local/bin/node', '/usr/local/bin/npm', 'start' ]
2 info using npm@5.6.0
3 info using node@v9.2.1
4 verbose run-script [ 'prestart', 'start', 'poststart' ]
5 info lifecycle magicmirror@2.1.0~prestart: magicmirror@2.1.0
6 info lifecycle magicmirror@2.1.0~start: magicmirror@2.1.0
7 verbose lifecycle magicmirror@2.1.0~start: unsafe-perm in lifecycle true
8 verbose lifecycle magicmirror@2.1.0~start: PATH: /usr/local/lib/node_modules/npm/node_modules/npm-lifecycle/node-gyp-bin:/home/pi/MagicMirro$
9 verbose lifecycle magicmirror@2.1.0~start: CWD: /home/pi/MagicMirror
10 silly lifecycle magicmirror@2.1.0~start: Args: [ '-c', 'electron js/electron.js' ]
11 silly lifecycle magicmirror@2.1.0~start: Returned: code: 1  signal: null
12 info lifecycle magicmirror@2.1.0~start: Failed to exec start script
13 verbose stack Error: magicmirror@2.1.0 start: `electron js/electron.js`
13 verbose stack Exit status 1
13 verbose stack     at EventEmitter.<anonymous> (/usr/local/lib/node_modules/npm/node_modules/npm-lifecycle/index.js:285:16)
13 verbose stack     at EventEmitter.emit (events.js:159:13)
13 verbose stack     at ChildProcess.<anonymous> (/usr/local/lib/node_modules/npm/node_modules/npm-lifecycle/lib/spawn.js:55:14)
13 verbose stack     at ChildProcess.emit (events.js:159:13)
13 verbose stack     at maybeClose (internal/child_process.js:943:16)
13 verbose stack     at Process.ChildProcess._handle.onexit (internal/child_process.js:220:5)
14 verbose pkgid magicmirror@2.1.0
15 verbose cwd /home/pi/MagicMirror
16 verbose Linux 4.9.59-v7+
17 verbose argv "/usr/local/bin/node" "/usr/local/bin/npm" "start"
18 verbose node v9.2.1
19 verbose npm  v5.6.0
20 error code ELIFECYCLE
21 error errno 1
22 error magicmirror@2.1.0 start: `electron js/electron.js`
22 error Exit status 1
23 error Failed at the magicmirror@2.1.0 start script.
23 error This is probably not a problem with npm. There is likely additional logging output above.
24 verbose exit [ 1, true ]

could it be, that my downgrade to 2.1.0 is to blame?

Update npm:

npm install -g npm

Update node (should update npm too):

sudo npm cache clean -f
sudo npm install -g n
sudo n stable

yes, its on a raspberry pi 3.

pi@MagicMirror:~ $ node -v
v6.12.2
pi@MagicMirror:~ $ npm -v
3.10.10

edit:

well, first it started to install something. i did it via ssh, so i changed to my browser to surf in the web while the raspberry is installing. When i looked back i just saw this message shown above.

node -v
npm -v