After I entered Klamath, my weapon disappeared and it doesn`t come back when I leave the town. So do I have to get a new weapon every time I visit a town?

Looks like either a bug or you dropped it accidentally. No weapons should disappear just by entering a town.
Or maybe you switched from the quick item slot with the weapon to the other.

Well, I didn`t switch slots and I didn`t put it down. It also doesn`t appear in the inventory. By the way, the weapon that I used in the Temple at the beginning did also disappear when I was in the tribal village.
Hm, so it`s likely a bug and it seems as if I can`t play the game, because every time I enter a village(town it seems that my weapon get`s removed by the game!!!
And because this game is so old it won`t get patched anymore. Damn, now I`m really angry!!!

Strange. Is it "vanilla" version or after some unofficial patches? Do you use any mods?
It's strongly recommended to use (at least) Killap's Unofficial Patch (a lot of bug fixes):
http://www.killap.net/
... but if you don't mind for additional content (originally intended by developers but cut off from final release; nothing fancy fan made crap) then use Restoration Project.

I guess your weapons were spears, did you accidentally throw them in combat ?

Does removing the weapon from the quick slot and putting it into your normal inventory before entering the town as
workaround help ?

I'd try a clean redownload and reinstall of the Fallout 2 GoG version into a different folder, without mods. If the problem still appears you could try installing Killap's or other mods.

I installed the normal GoG version. I didn`t throw my spear. Oh, and how can I access my inventory while I`m travelling?

You can't. You have to stop first.

How?

How to stop traveling? Just click at some point in the map. A green arrow will appear, just click inside.

Hm, which one should I download? The Unofficial Patch or the RP Patch?

You can download both... ;-)

"Unofficial Patch" is just a patch - it will fix many bugs and will not add nothing new (restored) content. If you want to experience "original" game without bugs then install it.

"Restoration Project" restores content made by developer but cut off from final game (for some reason like "we have no time nor money to do this"):
http://falloutmods.wikia.com/wiki/Killap's_Fallout_2_Restoration_Project

So, both patches work with the GoG version? I guess I have to start a new game, after applying the patch, right?

Of course.

Yes, both works with GOG.com version and yes it requires new gameplay.

just a note to people new to the first two Fallout games - PEOPLE STEAL YOUR STUFF! It was pretty well known in fallout 1 that in the Hub, there was a band of child thieves that would run around town. If the child thief ever stopped near your character, they would often remove a random item from your inventory.

when people figured this out, you could drop your inventory outside of town, arm a grenade in combat mode, then put the grenade back in your inventory - then, you'd wander around town until the child finally sidled up to you and stole it - watch them explode in red mist a few seconds later and laugh your ass off.

moral of the story is - the wasteland in fallout 1 and 2 is particularly cutthroat, and it's not impossible at all that an NPC managed to steal your weapon.