Juliet is the only surviving Capulet child; all of the previous potential heirs have met with tragedy, the "Earth hath swallowed them" (1.2.14), meaning they are in their graves. That these offspring were sons is clear. Capulet's "hopes" were for securing his property and money after his death. While he has a soft spot for Juliet and hopes that she will be happy, this happiness is by no means his primary motive.
There are mechanations going on here that require some knowledge of the period. Women were not allowed to own property or be the executors of anything in an estate. His daughter can only be the recipient of the family assets if she marries. Capulet's hope now rests on Juliet and making certain that she marries advantageously. That is, her husband must be elite and weathly. The match must be one of which he approves to carry on his line, even if it is through the lesser choice of having a daughter do so rather than a son.
Remeber that Juliet is only allowed to "choose" in the "scope of choice" that Capulet settles on for his daughter (line 18). His arch enemy's son is certainly not within this narrow field.
Yes, Capulet is talking to Paris in lines 13-33.
Capulet is telling Paris that all of his previous children have died in childbirth and that Juliet was the only one to survive. He tells Paris to go ahead and try to win Juliet's love because his approval for them to get married is only half of the battle. Paris can't just have Juliet to marry, Capulet wants Paris to actually get Juliet to want to marry him. Only if Juliet agrees will Capulet move forward with the marriage.
Then Capulet invites Paris to a masquerade ball. Capulet tells him there will be many beautiful young women there and he won't know which one is Juliet. Capulet tells Paris that he is to check out all of the lovely ladies and if the one he likes best is Juliet, that is fine, but she might not end up being his favorite.