Let's say I've got a 802.11n WiFi network in which all hosts are using 802.11n standard. If you start sniffing with an 802.11b/g card (using airodump for instance) will you be able to perform any malicious operation on my network?
802.11 b/g cards can only sniff 802.11b/g traffic. They cannot see 802.11n traffic. See caveats A & B, below.
Caveat A: Many 802.11n APs are configured to allow b/g connections. Once you have an IP connection, all kinds of things are possible, including ARP cache poisoning, mac-floods, MITM attacks, etc...
EDIT for queston in the comments:
Caveat B: Even if the sender / receiver have 802.11n radios, and the attacker only has 802.11b/g, it's very possible to direct RF jamming signals towards an 802.11n sender / receiver and make them intentionally fall back to 802.11b/g data rates.
Quoting Cisco's 20 Myths of Wi-Fi interference (emphasis mine):
... non-802.11 types of interference typically don't work cooperatively with 802.11 devices, and can cause significant loss of throughput. In addition, they can cause secondary effects such as rate back-off, in which retransmissions caused by interference trick the 802.11 devices into thinking that they should use lower data rates than appropriate.
When an 802.11n PHY falls back to 802.11b/g rates, it changes the physical layer signalling from 802.11n to 802.11b/g (per-client).
You will not be able to capture any traffic using the high throughput (MCS) data rates. However, by default 802.11n devices will support and use legacy data rates depending on the situation.
As long as the AP and/or client are using legacy data rates, you should be able to capture their traffic. This would include most (if not all) management and control frames which will typically use a lower/legacy data rate. This is opposed to data frames which run at the best possible data rate, including the EAPOL exchange of keying material. So you will miss that unless the AP and client both use a legacy data rate for this exchange.
So in answer, yes there are many malicious operations that can be carried out on an 802.11n network by a legacy client. In particular anything that would exploit management and/or control frames, plus any data traffic that uses legacy data rates.