IMAP Email Client for Blackberry

Most people, including me, use the Blackberry built-in email client to read their messages. It is a great email client in many ways and I am very happy with it, but there are two annoyances:

1. If I delete an email on IMAP server from my desktop email client such as Thunderbird, it is not automatically deleted from my Blackberry email client. (On the other hand, if I delete an email in my Blackberry email client, I can optionally request the message to be deleted from the IMAP server).
2. It takes between 5 - 30 minutes for new mails to be pushed to my Blackberry. Actually this is not so bad if the interval between pushes is somewhat predictable, but it appears to be random to me (perhaps it is determined based on system load)?
   

Note that I do not have an Blackberry Enterprise Server deployed in my workplace, which may not have the above limitation. I am just using the Blackberry data plan / Blackberry Internet Service (BIS).

To work around problem #2 in particular, I sometimes use an IMAP email client to check if I have new emails from my mail server. The best, free one I found so far is LogicMail. It supports both POP and IMAP including SSL. You can see some screenshots from here.