BB, HMQ can remove the PM here if she so chooses. The right of the monarch to do this has never been rescinded, but the last time it was exercised was 1834, when William IV sacked Lord Melbourne [alluded to in the wiki article], on the grounds that he thought Melbourne was too kind to Catholics (despite the fact that Catholics had been emancipated in 1829).
This was regarded as a gross mis-use of power by the monarch, and it became accepted that the monarch would never sack the PM, unless (for instance) the PM went totally bonkers and, despite being deposed by his party as a result, refused to leave number 10, claiming he was still the legitimate PM. If HMQ then said ' one begs to differ, as one believes Sir Alan Sugar says "you're fired", Prime Minister' that would be that. Unless there were the exceptional circumstances noted above, this would now create a huge constitutional crisis, which - almost certainly - parliament would probably win.
As for dissolving parliament, she has done for every general election since 1955. The monarch is the only person who can dissolve parliament.
Again, it would take exceptional circumstances for HMQ to do this - most notably if a parliament declared that it was going to ignore the requirement for an election every five years and continue on its merry way. What happens normally is that the PM decides when he/she is going to have the general election and goes to the palace to ask HMQ to dissolve parliament. A royal proclamation is issued calling for the issuing of the writs to summon a new parliament.
Conversely, the monarch may refuse the request for a dissolution - for instance, if a minority government requested a dissolution, but it was clear that a coalition commanding a majority could be formed, the Queen might (upon taking advice) refuse, allow the coalition to form, and then invite the leader of the coalition to become PM.
We now return you to the question of whether Mr Bliar should be saluted in the approved manner, or in a less orthodox fashion...