Unfortunately, you cannot use GMail to talk IMAP to another account. So you cannot easily keep the structure from your existing account without external help.
However, you can use POP3 which your IMAP account probably supports. POP3 can be used to pull in email from another account but it doesn't know about folders so you can only really pull in mail from the INBOX.
It is also easy to set up Google to allow sending as a different user. But some email domains (in fact many these days) do not allow email to be send from anywhere but their own SMTP service. This is to help reduce SPAM. So that might not work, you will have to try it. When you send an email, you will be able to choose the sender address if you have configured any additional ones.
Either way, the settings for both are found under the gear icon in GMail, under Settings, Accounts and Import.
Note also, that your GMail address will be included in the "Sender" header of the mail so knowledgeable people will be able to find it.
OK, so that is the GMail only way and it doesn't completely answer your questions.
Alternatively, you can use one of a couple of methods with external help:
Use an email client to copy/paste (or drag/drop) existing IMAP folders to GMail folders that you create. In Thunderbird, Outlook or whatever, you can access GMail via IMAP and the tags in GMail are treated as folders.
Or, you could use a mail sync client such as OfflineImap which is a command-line, Python-based tool for synchronising IMAP folders.