Hello Tamar,

No,

1) When you use sendmail from the command line, the message is built with local information: the sender, as well as the sending host (MailUserAgent part of sendmail). Next the MAU passes the email message on to the MailTransferAgent part of sendmail, still on the same system. So it will in effexct act as a relay. So for a message to get proper credentials (meaning it will not be rejected by the receiving mail server or any mailserver in between) the local sendmail.cf configuration should be in order.
2)a) what do you mean with "my smtp server", the sending system, which is a smtp server, the next hop from the sending machine (aka relay server), or the receiving mailserver (or that companies mail gateway)
2)b) assuming (dangerous, but therfor I state the I assume) it is the receiving mail (gateway) server, it is nice that you can ping, but this only indicates that there is a physical connection possible (level 0 and 1 of the OSI model) rejecting the email message is a little bit more sophisticated, and happens on aplication level.

The acceptance criteria in place nowadays include the validation (!!) of the sending machine and sender. A corporate mail system will reject an incomming email, when it cannot find a valid MX record for the "sending_domain" (here I mean with sending_domain, either companies mail_domain, e.g. hp.com, or the actual sending mailserver, e.g. smtpx.bbn.hp.com), or sender (e.g. I send from inside the hp.com network domain, as if I send from my yahoo.com domain, since HP and Yahoo! have no email relationship, it is unlikely, that mail originating from Yahoo! will be sent through hp, and therefor will be rejected).

To make a long story longer:
If you want your email message to be recieved, you have to create it in such a manner, that it will be accepted. Most of the times it will be silently rejected, when incorrectly composed, so that spammers do not learn from anti-spamming rules. If you are friends with the receiving side, you might ask for their log file, and see what is wrong with your message.

Often the sending system is not MX resolvable (nslookup -q=mx host/domain_under_investigation) or the sender (From:) does not exist, or has an invalid "domain" (user@system.domain, where system.domain has no MX record)

HTH