The House of Lords serves primarily as a constitutional safeguard, scrutinizing and amending bills that have been passed by the Commons. If it weren't for them then there'd be nothing to stop the Commons from, for example, depriving all gingers of the vote if that's what the public were in favour of. It can't block a bill completely, and so doesn't threaten the democratic process, but it can force people to think again about rash decisions. And, since it's unelected, its members are less likely to be partisan and more likely to actually consider whether or not a bill would be good for the country.
As for the Queen, her signing a bill into law is little more than a formality these days. The last monarch to withhold Royal Assent was Queen Anne in 1708. And, since the Lords would already have dealt with any particularly extreme bills, there's no need for Elizabeth to ever do the same. Nevertheless she could still have an important part to play if there is ever a constitutional crisis such as a military coup or a terrorist attack on Parliament. It's happened before in other countries - King Juan Carlos of Spain used his power to rebuild the country after Franco died, and then single-handedly crushed a coup d'etat by refusing to recognise its leaders. An elected president would never have been able to do that in such a volatile political environment.
The monarchy part's a tradition - a rubber stamp - she'd never decline assent. I believe the last time a monarch tried to refuse was in 1707 to try and prevent the formation of the Union.
The House of Lords is also a tradition, although it still performs a vital function today in scrutinising legislation, proposing amendments to legislation and acting as a barrier against governments acting arbitrarily. Their power's limited in that all amendments have to be accepted by the Commons and they can only delay leglislation for up to a year. But they have qualities that are pretty vital - politically independent crossbenchers, expertise in the form of lawyers, ex-politicians, businessmen (etc.).
The House of Lords cannot stop something becoming law if the Commons is sufficiently determined. Its purpose is to put a brake on hastily considered matters and invite second thoughts. It knows and accepts that the Commons has the right to ignore its concerns but then the Commons is also held exclusively to account for such a decision.
So their purpose, essentially, is as an improver of legislation.
The monarchy's purpose is largely ceremonial nowadays but the monarch possesses certain powers that are only ever used in emergencies, and are vested in the monarch to avoid any temptation of elected officials to use them more frequently.
The House of Lords is partly there to scrutinize the Commons (although its effectiveness in doing this, since it is unelected and therefore not as democratically legitimate as the Commons, is limited) and partly there to provide a second opinion on Bills. The HL is meant to be critical of bills - to point out ways in which they could be improved and to provide expertise in relation to bills. It has been argued that the HL is best unelected for this reason because if the Second Chamber were elected then its ability to be impartial and focus on its scrutinising role would be impaired, because its members would be focusing more on getting themselves reelected. But your point about democracy is certainly an important one, OP.
It is for this reason that Parliament is currently considering making part of the Lords elected. But the plan is to keep part of it unelected because of the expertise/impartiality element.
The HL used to be the final court of appeal in England as well but this was changed under the Constitutional Reform Act 2005 which created the Supreme Court in 2009. It didn't seem right to have a parliamentary body whose members also adjudicated in appeal cases.
As for the monarchy, it is a constitutional convention (not quite law, but a strongly held tradition) that the Queen will accept decisions made by Parliament. It would probably be a political outrage if she were to disagree.
The Lord's are not elected- they used to be hereditary but now they are appointed by the government and serve for their lifetimes. Lord's are basically used as a check on government power, since the Commons is ill placed to do this as its usually made up of a single majority party that also makes up the government- party unity and party discipline means that all the MPs in the Commons that belong to the governing party will most likely vote on the party line. The Lords sees itself as the guardian of the mandate- meaning that if a government tries to pass a bill that they didn't mention in their electoral manifesto, the Lords will block it if they disagree.
But, yes, you are right, it's not very democratic which is why pretty much all parties are now in favour of Lords reform- they want some form of elected second chamber. However, this issue is very controversial as an elected Lords would make it much more assertive and therefore make Parliament less efficient. Also, they can't agree on what percentage of peers should be elected or appointed, and since its a pretty huge constitutional change they do need some degree of unity on the decision.
Also, the Lords can initiate legislation if it wants, and it used to be the final court of appeal in the UK until the Supreme Court in 2009. There is still a significant amount of legal expertise in the Lords however and they can be used as a source of advise.
As for the monarchy, it doesn't have much power any more, and hasn't really since the glorious revolution. The monarch is officially the Head of State, though in practice most of the traditional duties of the monarch are now exercised by the Prime Minister (royal prerogative). The PM is Head of government- if he was also head of state he would be a President (if elected and accountable to the people), or a dictator or absolutist monarch. The monarch does have to give royal assent to bills passed by Parliament before they become law but this hasn't been refused since the 1700s. The monarchy is seen as the "dignified" part of our constitution, basically the monarch only has ceremonial powers, but was traditionally seen as a symbol of political unity, who promoted popular allegiance.