Adding music to my iPhone from a new computer? Is it possible to do so without erasing music already copied?

Question

I have an iPhone 5c. I originally set it up on my computer at home and copied my music over using iTunes.
I am now away on a trip for a few weeks with work and have a work laptop. I have downloaded iTunes and authorized the laptop. I have some music I want to add it to my iPhone however when I try and manually move the music to the iPhone icon in iTunes I get a little red "no entry" symbol show up.
I understand I can select "manually manage music and videos" however when I click this it wants me to delete all my music and videos and only sync the music already wthin iTunes on this laptop. Obviously this means I will lose all the music currently on my iPhone and only copy the recently downloaded music to it.
1) Is it possible to copy the new music without it deleting my previous music in my iPhone? I don't want to lose my original 50 albums just so I can add another 3!
2) Will syncing it also delete my apps and save game data etc? Or will it only affect my music and videos?
3) Is the "manually manage music and videos" a iPhone option or a iTunes option? I am almost certain I set it originally on my PC so the fact it is now unticked confuses me.
Thanking you in anticipation...

Answer 1

The aim is prevent casual copying so you can't go around to a friend's house and simply copy all of the media from your device onto your friend's computer, or vice versa.
There are tools in the user tip that can recover all of the non-iTunes-purchased media, however the process is really a last restort for when the original library has been lost or damaged and there is no backup.
I believe it is only the media that is at risk, provided you backup the device and transfer apps, but I could be wrong.
FWIW you can, and should, backup your library on a regular basis. If your library is in a portable form you can backup to a portable drive that you can then take with you and manage the device from any other computer running the same build of iTunes.
1) Is it possible to copy the new music without it deleting my previous music in my iPhone? I don't want to lose my original 50 albums just so I can add another 3!
Not when using a different library than was used to put the first 50 there.
2) Will syncing it also delete my apps and save game data etc? Or will it only affect my music and videos?
Uncertain. If you backup the device and transfer purchases then the apps and saved should be safe, and could be restored if needed.
3) Is the "manually manage music and videos" a iPhone option or a iTunes option? I am almost certain I set it originally on my PC so the fact it is now unticked confuses me
iPhones may only be manually managed with a single library. Other devices can be manually managed from multiple libraries. I have no idea why this arbitrary restriction is there, but there is a workaround as given above. You can clone your library to another drive or computer and it will work fine.
tt2