Actually, that’s a flail. A mace is just a glorified club. Pictures here.
On preview, seaworthy beat me to it, but I have a link. 
The House of Commons Mace is a much more impressive object.
qts is right. The Prime Minister is not the leader of the largest seat-winning party. He/She is the individual who can count on the largest support amongst all other MPs in a way that will form a working government. In practice, however, this usually is the leader of the largest party.
But there is nothing to stop absolutely any MP going to the Queen and telling her he/she has the most support among the MPs and intends to form the next government. The Queen’s not going to give them her blessing if it’s just wishful thinking though. Mind you, lack of her blessing wouldn’t stop a government forming either. It just goes to show that the whole UK method of government is based on nothing other than convention and practice. “The done thing.” If anyone wanted to be awkward about it they could ride roughshod through the whole arrangement.
Damn. A MP tried to swing that in protest but broke it instead? I bet everybody was pretty pissed about that.
It’s my understanding that the prime minister need not be a member of the House of Commons. Is that right?
Many Prime Ministers came from the House of Lords before the 20th century.
It could conceivably happen that the Prime Minister wasn’t elected in his constituency, but was still the leader of the party with the most seats (or leader of a coalition). For instance in Canada this might happen if the NDP got a majority government but Jack Layton (leader of the NDP) didn’t win his riding.
Maybe, but it has long since been unacceptable for the British Prime Minister not to be a member of the Commons. When the Earl of Home (pronounced ‘Hume’) became PM at short notice in 1963, he resigned his peerage, becoming plain Sir Alec Douglas-Home, and a Tory MP in a safe seat fell on his sword to allow Douglas-Home to contest the resulting by-election.
Mace-waving is rare, but Members getting suspended for other reasons happens every couple of years. Seems that during Bernard Weatherill’s 1983-92 stint as Speaker, four Members were “named” (a technical term in this context) and then suspended: Tam Dalyell (twice), Dennis Skinner, Martin Flannery and Brian Sedgemore. Typically, a suspension only lasts a few weeks.
To be pedantic, not quite: George Younger only stood down as the candidate in the Kinross and West Perthshire by-election so that Douglas-Home could stand. Younger, of course and entirely uncoincidentally, did go on to become an MP (for Ayr) anyway.