Lots if good advice above.
Re mounting: dogs do this, also females to females, males to males, females to males, for many reasons and why is not well understood. It can be play behaviour, it can be one dog showing the other it runs the show, it can just feel good, it can be sexual interest. Desexed animals are not asexual animals remember.
So he may just find her an attractive older woman. :lol: There's a thread on mounting behaviour in the Library section that has a lot of info on what experts think this kind of behaviour might be all about.
Re the weeing: a very, very common reaction for a dog that is a bit stressed or also, simply identifying its territory or claiming, this is my home (that's one reason they often pee on our beds!). My mom's dog does this when she is stressed and did it for a while when she first went to live with my parents. Lily did it too, and Jaspar would pee on the floor for a while right after I got Leo. So just keep doing what you did -- I am sure it will quickly stop and may not be repeated at all.
Prince is just a bit grumpy because he doesn't yet realise how much fun it will be to have a friend.
Think of a child whose cousin has come to stay for a visit -- if they aren't friends, the child might be a bit put out at not getting 100% attention and also that his routine is upset.
Lucy is looking for lots of attention because she wants reassurance and also is stressed at being in a new situation where nothing is familiar. Just ignore her constant demands and treat both dogs as you would prefer to keep on -- eg don;t give special privileges to either if that's not what you want to keep on doing. But do make sure Prince gets extra time.
I wouldn't focus on letting him out first or anything like that -- this can actually cause problems if the natural hierarchy these dogs choose ends up that she is the more dominant. Stressing the he is more important can throw the 'real boss' into bad behaviour as she starts to try and prove through unwanted behaviour that hey, it's ME that is top dog! Neither dog is likely to care who is 1 and who is 2 (and it often is the female who is 1). They just like to be aware of it for their own relationship, just as you would know in your place of work who to defer to and who is your work equal and who can be asked to do things for you. When we intervene because we feel the first dog in the house should be number 1, and that's not the actual case, it can make the number 2 dog anxious for being pushed into a role it doesn't have, and the number 1 dog anxious for being forced to have to prove to you and the other dog that it is the real boss.
Hence I'd just take a step back and not worry about those kinds of details. Just enjoy both, don't even worry if she snaps or he snaps, as long as they aren't actually fighting. A snap is just a polite but firm way for a dog to indicate someone or something is pushing its boundaries. They rarely actually make any contact and the other dog almost always will recognise the warning for what it is and desist and be more carefl in future.