Is there any way through DOSBox to do multiplayer with HoMM? I am thinking perhaps the 'network' option.

You answered your own question.

At least for HoMM3, Hamachi does the job (through VPN).

Not really.
I, for instance, was able to play through network only after manually connecting two DOSBox sessions.

Well, I get an error about netbios not being on. So, it an't be that easy, which is why I asked here.

Netbios should launch automatically. Do you run HoMM via shortcut or manually in DOSBox?
Also, did you install DOSBox with HoMM, or had one already?

I'm not actually using GOGs copy. I'm using a copy from "Heroes of Might and Magic Compendium ". Just copied directly from the disc.

[First I apologise if anything below makes no sense; maybe there's far shorter solution, but this is how I got HoMM to work in multiplayer :) ]
Ok, then:
0) add DOSBox to firewall exceptions, if you have one
1) download [url=ftp://ftp.ipswitch.com/ipswitch/Product_Support/Catipult/netbios.exe]netbios[/url] and put it into DOSBox directory
2) configure your dosbox.conf - under [IPX] section make entries:
Enable=1
Connection=1
ipx=true
3a) now, I don't know how Compendium version works (do you need a mounted image or whatnot), but before launching 'heroes.exe' enter commands:
#go to DOSBox directory
IPXNET STARTSERVER
netbios.exe
#go to HoMM directory & launch a game
now for the tricky part:
STARTSERVER command makes your computer a host; on the computer you want to play with (a guest), do all steps above but replace that command with:
IPXNET CONNECT host's IP number
now you should be able to connect through 'network';

I have Windows 10 and it won't help me much. I have gog.com version. When I push Network button then the game throws me to the desktop and that is it.

I am also using Windows 10 and running into the issue where in never connects to the guest. Is there a video or something that might help?

EDIT: I got multiplayer to work. There are one of two conditions that needed to be met either 1.) you play via local wifi or 2.) use a VPN to simulate it.

Actually you should either have a VPN; or a static non-NAT'ed IP with a properly forwarded through the firewall port on the server side.
VPN is more flexible, more safe and easier to setup, therefore it is recommended. Direct connection via forwarded port might give lower ping values and more stable connection, though.

Can you transfer gold and other resources between players in Heroes 1 MP, and is there a team battle feature?

Same for me, but it works if you go to Start -> Programs -> Heroes of Might and Magic [GOG.com] -> Launch Multiplayer

Then, on on computer, select "Start server" and on the other computer, do the same thing and choose "Start client", then enter the IP address of the server (note you probably have to disable firewall on the server). Then it will load the game and you can now choose Multiplayer, then Network as the option. Great work GoG.com!!