Lex: That's a nice story. But why is it so important to you really?

Lana: My parents met here. I guess I'm just looking  

for something to hold onto. (sth where she could find support in?)Yes. Something permanent in a changing world.

Lex: So I'm supposed to preserve this monument to (This to bothers me. To preserve it for/because of her?)

Lana Lang's sentimental memories?If I build a monument because I want to honour and remember Tom, we say 'I built a monument to Tom'.
Lex is saying that Lana thinks he is supposed to preserve a monument to Lana's memories, ie he is supposed to honour and remember them. Lex says this sarcastically, because he thinks it's an unreasonable request.

Lana: That's not what I meant.

Lex: You're a teenager, Lana. Mercurial doesn't do you justice. (Is it sth to do with her mood?) 'Mercurial' means subject to quick changes. It's often used to describe a person's temperament.The underlined phrase means that the word 'mercurial' is not sufficient to describe you, ie you change your temperament very, very, very quickly.

Lana: I thought you might understand. Clearly, I was mistaken.

Lex: This is business, Lana. All these old storefronts are

becoming professional office space. This town needs a parking garage.

Lana: So you're going ahead with your plans?

Lex: You haven't given me any reason to change my mind. If you'll excuse me,

I have a pressing matter to attend to. (He has a very urgent meeting?) Not necessarily a meeting, but he has to deal with something that is urgent.

Best wishes, Clive

Madhulk

“ something to hold onto. (sth where she could find support in?)  Perhaps emotional support, perhaps something through which she can make a connection to the past  -  to hold in her memory.

preserve this monument to (This to bothers me. To preserve it for/because of her?)  This tribute to Lana's memories  -   like a memorial to someone  -   Ode To A Nightingale

 Mercurial doesn't do you justice. (Is it sth to do with her mood?) Yes.  There are so many different attributes, alluding to the planet (astrology), the god, the element  -  It's hard to know which one he's referring to.   probably "eloquence" in this case.  (It doesn't become her, especially as a teenager.) 

Edit.  Clive's probably right.  I had rejected that as being typical for a teenager, but Clive's point about beyond Mercurial is good, if I understood correctly.

I have a pressing matter to attend to. (He has a very urgent meeting?) Could be.  (It's just an excuse to leave, right?)  Maybe he needs to visit the lou. ”