Tanya, thank you for your response. Please send my big Thanks to Evgeny as well for the job well done. She responded to me in a direct support e-mail.
These are the steps that worked for me (provided by SmartBear support)
1. Create a backup copy of your current project.
2. Open your project suite in TestComplete 9.31.
3. Double-click the NameMapping item in the Project Explorer panel.
4. Set the "Store Code Completion Information" option to "Do not store data for Code Completion" in the Name Mapping editor. Then, click Yes.
5. Set the "Store Code Completion Information" option to "Store brief data" in the Name Mapping editor.
6. Save the changes. Close TestComplete. Then, open it again and see whether it improves the situation.
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My result is from 7-8 minutes of load time, to less than a minute
TestComplete's memory consumption came down from 1+GB to 300MB
Also, my Namemapping file came down from 175MB to 2MB
And so far no more freeze on playback.
I've realized that this whole thing started when I mapped a part of the product under test which had a grid with 6000+rows and 20+ columns.
I've asked from some information on Namemapping and why it did that and I was given
http://support.smartbear.com/viewarticle/26988/Just a small suggestion: to include the information regarding namemapping "store brief" stuff to the article "Automated Test Performance Tips"
To SmartBear support. Thanks!