Basically, if our salary commitments are under the cap, we can only spend our cap space plus minimum contracts. However, we can also re-sign our own free agents regardless of how much cap space we have.
To prevent teams from using their cap space on a free agent like Nash and then turning around and signing their own free agents (because you don't need cap space for that), there is a rule that says if you want to be able to re-sign those free agents, they take up a certain amount of your cap even before they are signed. This amount is called a "cap hold."
In the case of players coming off their rookie contracts, the cap hold is defined as 250% of their previous year's salary - which is where that 7 million number comes from for Bayless. So until we sign him (or renounce his rights, in which case we need cap space to sign him) our cap space is taken up by that big fat cap hold.
The Nash contract will be around 10 million a season in all likelihood. As such, we will need 10 million in cap space to be able to sign him, as he is the Phoenix Suns' free agent, and not ours. As you can see in my breakdown above, we have only about 3 million because of Bayless' cap hold taking up all that space. To sign Nash, we will have to either let Bayless walk to another team, or clear some cap space (which is what the "amnesty" provision does - wipes one player off the team's books).
As for how much we can spend, teams can spend any amount they like. However, there are rules about when you can sign players. If you are under the cap, you have to use your cap space to sign free agents. If you are over the cap, you have to use exceptions (such as the Mid Level Exception, which allows a capped-out team to sign a player for 5 million a year; or such as the Bird Exception - which is the exception that allows a team to re-sign their own free agents as I discussed above) to sign players. As such, there is pretty much no way the Raptors
can get up to 70 million this year, as they will be under the cap to start the offseason. They will likely end up right around the cap level (58 million) give or take 5 million.