I find out that our country (slovakia) has lack of experience in mailing systems. There is no one who to ask and no discussion forums too.

I can do it on Forums like daniweb and other foreign sites. The problem is that the network in our country is not the same in functionality and in security too.
So there are things which has to be done in different way. This is what I want to find out here.

First question:

Do I need smarthosting?? (sendmail - what is it at all??)

can you describe, in detail, what you are trying to do?

I am working out a mail, http and file small server. It is on slackware linux with sendmail.
I had troubles with host name lookups with relaying.
I defined a smarthost for me. But there was no need. I found on other discussion forums an explanation.
The smarthost is needed when our mail exchanger is not accessible directly but through ISP smarthost.

what kind of troubles? lets address each issue on its own, instead of jumping into conclusions, ok?
so what problems exactly did you have, and what is your setup like there?

I was trying to configure sendmail with LOGIN and TLS support.
When I configure everything. I can send mail to internal mailboxes. I can receive mail from anywhere.
But I could't send mail to external recipient:
from user@mydomain.com to user@destdomain.com

I get an error:
stat=Deferred: Name Server: destdomain.com : host name lookup failure

ifconfig
cat /etc/resolv.conf
nmap localhost
dig mx yourdomain.com

also post the full smtp session printout

I was already able to resolve the problem.
I had errors in DNS zone definitions and named.conf.

I will post all the details but I don't have access to the server right now.

But even though I make the printout of the commands you posted. (If somebody has the same problem)
I can also make some short howto for sendmail config on slackware 12 on daniweb. If it will be usefull or good for the community.

glad to hear you resolved it.

as you probably realise now, the internet is international, and it doesn't matter where you are, it is the same everywhere that was referring to the network functionality in your country