A bit of background:
Thor's Hammer, Mjolnir, does not have the power to open the bifrost, it was smithed by the dwarves at Odins command. It was not the kings weapon and did not have the power or authority to open the bifrost bestowed upon it by Odin, in fact, Odin has prohibited Thor from using the bifrost on several occasions and giving him the ability to activate it himself might be considered counter-productive.
Heimdall(or Heimdallr) wields Hofund, also called the bifrost sword, which has the power to open the bifrost, and was given to him when Odin made him the keeper of the bifrost, as a key he could use to open it.
Odins spear, Gungnir is the weapon of the king, and is able to open the bifrost as well, as the weapon of the king and carries his authority.
Now onto the axe:
When Eitri introduces us to the mould of Stormbreaker he mentions that it "is a weapon for a king", and as the weapon of the king of Asgard, Stormbreaker is capable of opening the bifrost.
Your second point: Heimdall and Thor are able to access the bifrost while not in Asgard, which was possible but not shown onscreen in previous movies. At the end of Ragnarok, Asgard is destroyed, and the bridge with it. Similarly, the Bridge was destroyed in Thor, making it impossible for Thor to return to earth. He still manages to get there in time for Avengers, thanks to the All-Father (Odin)'s use of dark magic to activate the bifrost and transport him there (The official Thor: The Dark World prelude comic goes in to this in more detail).
Heimdall is able to summon the bridge on the Asgardian ship by making a plea to the previous All-Fathers (Odin, Bor, etc...) using dark magic: "All-Fathers, let the dark magic flow through me one last time". Thor himself is the All-Father now, and once he has forged Stormbreaker, he can use the dark magic to open the bifrost himself.