It's hard to answer the question "What is the max speed of 5GHz in mbps?" in a general sense. In the context of 802.11 (Wi-Fi), "5GHz" is a radio frequency band -- a range of frequencies that you can use for things like Wi-Fi. A range of frequencies doesn't have a performance limit, other than the theoretical Shannon limit, but calculating the Shannon limit requires knowing the signal-to-noise ratio of the transmission, which varies.
So instead, we have to talk about things like the max speed specified for use in the 5GHz band in the latest 802.11 specs (600mbps), or what the latest equipment can do (450mbps), or what your particular AP can do (300mbps), or what's the best rate a cheap client—such as clients that are only 1x1:1, or don't support wide channels (HT40, 40MHz channels), or can't do Short Guard Intervals—can get when connected to your router (150mbps, 130mbps, and 65mbps are the max speeds for clients with certain limitations typical of cheap gear) and what actual throughput you can expect in any of those cases (about 50-60% of the above numbers).
802.11n has provisions for operation up to 600 mbps signaling using APs and clients that have 4 radio chains (supporting 4 spacial streams, sometimes denoted as "4x4:4"), but note that no one has shipped 4x4:4 equipment yet.
In late 2009, 3x3:3, 450mbps 802.11n equipment started hitting the market. Apple's late-2009 refresh of the AirPort Extreme and Time Capsule supported that. Apple's "Thunderbolt" refresh of the MacBook Pros and iMacs in spring 2011 also supported that.
Your Netgear WNDR3700 N600 is only a 2x2:2, 300 mbps product, so it's no faster than the earliest draft-N gear from late 2006. But at least it's simultaneous dual-band, which is nice. I think that simultaneous dual-band 2x2:2 gear generally hit the market in 2008.
Also beware that 802.11 has lots of overhead, so the rule of thumb is that your TCP throughput will only be about 50-60% of your signaling rate, and you only get the best signaling rates when you're relatively close to the AP with no interference.
As for Question 2 (Range), you'll be fine. You probably won't get your top signaling rate throughout your apartment, but you'll get decent speeds. Remember to leave all the data rates and technologies enabled on both bands so that your clients can choose the best bands and rates to use for best performance at range. That is, leave b/g/n enabled in 2.4GHz, and leave a/n enabled in 5GHz. Don't be tempted to force either band to "N-only" (or "G-only" or "A-only") because that actually just limits the options for your clients, making them less capable of providing the best rate at range.