When using Tor, if I download a file form a download-manager (on which Tor is not running) or if I check my emails using an email-clients, will this have any adverse effect on my anonymity? Will my real IP-address be exposed?
As long as what you are doing with your non-Tor applications and what you are doing in Tor is not related in anyway, it is safe.
But if they are related, for example if you are starting a download in your non-Tor application, for which the link/torrent you found on Tor (e.g. using Tor Browser), your current Tor session is deanonymized (IP address exposed). Likewise if you send a non-Tor mail to someone those mail address you just viewed in Tor, deanonymized.
Choose the green icons and "New Identity" in Tor Browser to create a new anonymous session. This will close all your current tabs.
Obviously, your non-Tor applications IP address is always exposed.
The existing answers cover the issue of cross-contamination between your internet use through Tor and outside it. It is definitely important that you need to be careful to not e.g. paste links obtained from clearnet into Tor Browser and vice versa.
But even with that care, your ISP (and anyone monitoring them) can still determine that you are using Tor, and that information could be correlated with monitored activity inside Tor to confirm a suspicion, if someone is looking. This is how the identity of Jeremy Hammond was confirmed. A common method of disconnecting Tor usage from a personal identity is to connect to Tor from a public place1, and it is here that additional care must be taken: if you simultaneously connect to personally-identifying services over the clearnet while there, all benefit of the public place is lost. Note that this might not even be a conscious act like browsing Facebook or Stack Exchange; your email client automatically checking for new mail would be sufficient.
tl;dr if your threat model involves evading the FBI, don't mix Torified-traffic and clearnet activity. (And if your threat model involves a larger adversary, even that may not help.)
1This assumes the public place cannot itself be correlated to the personal identity either historically or during the actual time-period of use, which is itself hard to ensure. Humans easily fall into existing patterns, even if they are actively trying to avoid them.
If I understand you right, you have Tor running on your computer. This can be either by using the Tor Browser Bundle or Tor as a stand-alone process. Now you are downloading a file with some download manager or check your mails.
When you have net set your clients (download manager, email client etc.) to use Tor, they will use your current Internet connection. So at least your provider will see what you are doing. Especially your real IP address will be exposed to your provider and to third parties (site where you download the file, mail provider etc.).
Your anonymity within Tor is usually not affected by 'outside' software like in the case you're describing. The software doesn't knwo about Tor and doesn't use it.