Thanks for your reply. I changed the email settings to use the ISP's mail server on port 587, and outbound emails are now working. This wasn't necessary for the previous ISP for some reason, even though they also blocked port 25.
For inbound, the ISP recommends port 110 or 995 with SSL, so looks like I will need to search the forums to see how to get that working (unless you have some quick pointers). In the meantime, I looked at the qpsmtpd/recent log. It shows there are some connections from the internet, but I don't think I like what they are:
2011-05-28 03:24:22.051685500 20440 dispatching DATA
2011-05-28 03:24:22.054253500 20440 354 go ahead
2011-05-28 03:24:22.331832500 20440 spooling message to disk
2011-05-28 03:24:22.676934500 20440 virus::clamav plugin (data_post): Changing permissions on file to permit scanner access
2011-05-28 03:24:22.704729500 20440 virus::clamav plugin (data_post): clamscan results: /var/spool/qpsmtpd/1306578262:20440:0: OK
2011-05-28 03:24:22.709820500 20440 logging::logterse plugin (queue): ` 93.87.201.65 93-87-201-65.dynamic.isp.telekom.rs moscow-trade.ru <dypokow@moscow-trade.ru> <some email account on my domain> queued <201105281223.41394.dypokow@moscow-trade.ru>
2011-05-28 03:24:22.718280500 20445 queue::qmail_2dqueue plugin (queue): (for 20440 ) Queuing qp 20445 to /var/qmail/bin/qmail-queue
2011-05-28 03:24:22.773503500 20440 250 Queued! 1306578262 qp 20445 <201105281223.41394.dypokow@moscow-trade.ru>
2011-05-28 03:24:22.998715500 20440 dispatching QUIT
2011-05-28 03:24:22.998722500 20440 221 #######.net closing connection. Have a wonderful day.
2011-05-28 03:24:22.998726500 20440 click, disconnecting
2011-05-28 03:24:23.316330500 4465 cleaning up after 20440
2011-05-28 08:28:13.911608500 23427 Accepted connection 0/40 from 90.183.115.33 / gw03.ecic.cz
2011-05-28 08:28:13.912333500 23427 Connection from gw03.ecic.cz [90.183.115.33]
2011-05-28 08:28:13.927615500 23427 tls plugin (init): ciphers: ALL:!aNULL:!ADH:!eNULL:!LOW:!EXP:RC4+RSA:+HIGH:+MEDIUM
2011-05-28 08:28:14.982999500 23427 check_earlytalker plugin (connect): remote host said nothing spontaneous, proceeding
2011-05-28 08:28:14.993237500 23427 220 benten.drifand.net ESMTP
2011-05-28 08:28:15.427383500 23427 dispatching HELO gw03.ecic.cz
2011-05-28 08:28:15.433827500 23427 250 #########.net Hi gw03.ecic.cz [90.183.115.33]; I am so happy to meet you.
2011-05-28 08:28:15.927704500 23427 dispatching MAIL FROM:<amarkova@jnelectric.com>
2011-05-28 08:28:15.928993500 23427 full from_parameter: FROM:<amarkova@jnelectric.com>
2011-05-28 08:28:15.942568500 23427 getting mail from <amarkova@jnelectric.com>
2011-05-28 08:28:15.943002500 23427 250 <amarkova@jnelectric.com>, sender OK - how exciting to get mail from you!
2011-05-28 08:28:16.428201500 23427 dispatching RCPT TO:<different_user@mydomain.net>
2011-05-28 08:28:16.438454500 23427 check_goodrcptto plugin (rcpt): stripping '-' extensions
2011-05-28 08:28:16.467805500 23427 250 <different_user@mydomain.net>, recipient ok
2011-05-28 08:28:16.925829500 23427 dispatching DATA
2011-05-28 08:28:16.928363500 23427 354 go ahead
2011-05-28 08:28:17.429429500 23427 spooling message to disk
2011-05-28 08:28:17.503068500 23427 virus::clamav plugin (data_post): Changing permissions on file to permit scanner access
2011-05-28 08:28:17.527750500 23427 virus::clamav plugin (data_post): clamscan results: /var/spool/qpsmtpd/1306596497:23427:0: OK
2011-05-28 08:28:17.532635500 23427 logging::logterse plugin (queue): ` 90.183.115.33 gw03.ecic.cz gw03.ecic.cz <amarkova@jnelectric.com> <different_user@mydomain.net> queued <000a01c40b73$c2b5dea0$507b2a59@stevef577bc085dprll>
2011-05-28 08:28:17.540641500 23433 queue::qmail_2dqueue plugin (queue): (for 23427 ) Queuing qp 23433 to /var/qmail/bin/qmail-queue
2011-05-28 08:28:17.600063500 23427 250 Queued! 1306596497 qp 23433 <000a01c40b73$c2b5dea0$507b2a59@stevef577bc085dprll>
2011-05-28 08:28:17.932333500 23427 dispatching QUIT
2011-05-28 08:28:17.932341500 23427 221 #######.net closing connection. Have a wonderful day.
2011-05-28 08:28:17.932346500 23427 click, disconnecting
2011-05-28 08:28:18.935744500 4465 cleaning up after 23427
2011-05-28 10:56:44.811590500 4474 tls plugin (init): ciphers: ALL:!aNULL:!ADH:!eNULL:!LOW:!EXP:RC4+RSA:+HIGH:+MEDIUM
2011-05-28 10:56:44.966383500 4474 Listening on 0.0.0.0:25
2011-05-28 10:56:44.966390500 4474 Running as user qpsmtpd, group qpsmtpd
2011-05-28 10:56:44.966394500 4474 Initializing spool_dir
2011-05-28 10:56:44.966398500 4474 Permissions on spool_dir /var/spool/qpsmtpd/ are not 0700
2011-05-28 10:56:44.966402500 4474 size_threshold set to 0
2011-05-28 16:41:58.172947500 4476 tls plugin (init): ciphers: ALL:!aNULL:!ADH:!eNULL:!LOW:!EXP:RC4+RSA:+HIGH:+MEDIUM
2011-05-28 16:41:58.242911500 4476 Listening on 0.0.0.0:25
2011-05-28 16:41:58.242919500 4476 Running as user qpsmtpd, group qpsmtpd
2011-05-28 16:41:58.242923500 4476 Initializing spool_dir
2011-05-28 16:41:58.242927500 4476 Permissions on spool_dir /var/spool/qpsmtpd/ are not 0700
2011-05-28 16:41:58.242931500 4476 size_threshold set to 0
2011-05-29 12:36:33.111213500 4479 tls plugin (init): ciphers: ALL:!aNULL:!ADH:!eNULL:!LOW:!EXP:RC4+RSA:+HIGH:+MEDIUM
2011-05-29 12:36:33.325379500 4479 Listening on 0.0.0.0:25
2011-05-29 12:36:33.326838500 4479 Running as user qpsmtpd, group qpsmtpd
2011-05-29 12:36:33.327185500 4479 Initializing spool_dir
2011-05-29 12:36:33.329976500 4479 Permissions on spool_dir /var/spool/qpsmtpd/ are not 0700
2011-05-29 12:36:33.330862500 4479 size_threshold set to 0
2011-05-30 14:23:49.216612500 4479 tls plugin (init): ciphers: ALL:!aNULL:!ADH:!eNULL:!LOW:!EXP:RC4+RSA:+HIGH:+MEDIUM
2011-05-30 14:23:49.224640500 4479 Initializing spool_dir
2011-05-30 14:23:49.224646500 4479 Permissions on spool_dir /var/spool/qpsmtpd/ are not 0700
2011-05-30 14:23:49.224651500 4479 size_threshold set to 0
2011-05-30 15:14:31.033047500 4479 tls plugin (init): ciphers: ALL:!aNULL:!ADH:!eNULL:!LOW:!EXP:RC4+RSA:+HIGH:+MEDIUM
2011-05-30 15:14:31.040117500 4479 Initializing spool_dir
2011-05-30 15:14:31.040123500 4479 Permissions on spool_dir /var/spool/qpsmtpd/ are not 0700
2011-05-30 15:14:31.040128500 4479 size_threshold set to 0
2011-05-30 15:39:04.896941500 4479 tls plugin (init): ciphers: ALL:!aNULL:!ADH:!eNULL:!LOW:!EXP:RC4+RSA:+HIGH:+MEDIUM
2011-05-30 15:39:04.896951500 4479 Initializing spool_dir
2011-05-30 15:39:04.900205500 4479 Permissions on spool_dir /var/spool/qpsmtpd/ are not 0700
2011-05-30 15:39:04.900212500 4479 size_threshold set to 0
2011-05-30 17:56:37.499279500 4479 tls plugin (init): ciphers: ALL:!aNULL:!ADH:!eNULL:!LOW:!EXP:RC4+RSA:+HIGH:+MEDIUM
2011-05-30 17:56:37.505181500 4479 Initializing spool_dir
2011-05-30 17:56:37.507777500 4479 Permissions on spool_dir /var/spool/qpsmtpd/ are not 0700
2011-05-30 17:56:37.508568500 4479 size_threshold set to 0
2011-06-12 19:33:09.752967500 4479 tls plugin (init): ciphers: ALL:!aNULL:!ADH:!eNULL:!LOW:!EXP:RC4+RSA:+HIGH:+MEDIUM
2011-06-12 19:33:09.822195500 4479 Initializing spool_dir
2011-06-12 19:33:09.822202500 4479 Permissions on spool_dir /var/spool/qpsmtpd/ are not 0700
2011-06-12 19:33:09.822207500 4479 size_threshold set to 0
2011-06-12 22:46:35.388267500 4474 tls plugin (init): ciphers: ALL:!aNULL:!ADH:!eNULL:!LOW:!EXP:RC4+RSA:+HIGH:+MEDIUM
2011-06-12 22:46:35.443557500 4474 Listening on 0.0.0.0:25
2011-06-12 22:46:35.443565500 4474 Running as user qpsmtpd, group qpsmtpd
2011-06-12 22:46:35.443569500 4474 Initializing spool_dir
2011-06-12 22:46:35.443572500 4474 Permissions on spool_dir /var/spool/qpsmtpd/ are not 0700
2011-06-12 22:46:35.443577500 4474 size_threshold set to 0
2011-06-13 22:53:47.023284500 4474 tls plugin (init): ciphers: ALL:!aNULL:!ADH:!eNULL:!LOW:!EXP:RC4+RSA:+HIGH:+MEDIUM
2011-06-13 22:53:47.028060500 4474 Initializing spool_dir
2011-06-13 22:53:47.028066500 4474 Permissions on spool_dir /var/spool/qpsmtpd/ are not 0700
2011-06-13 22:53:47.028070500 4474 size_threshold set to 0
What does it mean by "Changing permissions on file to permit scanner access"? Also it looks like it's trying to send email from one account to another.