I went a bit ahead in the investigation of this, but then got stuck again...
So, I found out, that saphostexec wasn't running at all! I found out that its profile file host_profile had completely different contents than usual. Then I found an older profile, renamed it to the actual profile and then started saphostexec. A few minutes later it died!
I started investigating and I found the following error(s) in the log file dev_saphostexec:
[Thr 01] Fri Dec 12 12:03:33 2014
[Thr 01] *** ERROR => Connection of channel 5 failed [ServerChanne 67]
[Thr 01] Fri Dec 12 12:03:34 2014
[Thr 01] *** ERROR => Connection of channel 6 failed [ServerChanne 67]
[Thr 01] *** ERROR => Connection of channel 7 failed [ServerChanne 67]
[Thr 01] Fri Dec 12 12:03:35 2014
[Thr 01] *** ERROR => Connection of channel 8 failed [ServerChanne 67]
[Thr 01] *** ERROR => Connection of channel 9 failed [ServerChanne 67]
[Thr 01] Fri Dec 12 12:03:36 2014
[Thr 01] *** ERROR => Connection of channel 10 failed [ServerChanne 67]
[Thr 01] *** ERROR => Setup of too many communication channels failed. Giving up. [ServerChanne 98]
[Thr 01] *** ERROR => Channel Accept failed [HostExecServ 242]
[Thr 01] *** ERROR => Start of HostService failed
[Logger.cpp 391]
A quick google search returned the following troubleshooting guide:
SAP Host Agent Troubleshooting Guide
I have tried everything that is recommended in the relevant section for my problem. I have checked using the touch command, under the sapadm user, that he is able to write all the described locations and when I tried to start saphostexec and it died again a few minutes later with exactly the same errors in the log file, as cited above.
So, this is really strange and I would really be very happy if someone helps me to move forward!