What are some ways to predict that a double replacement reaction will occur?
To determine if a double replacement reaction will occur, you need to look at the products. You correctly balanced the equation and flipped the anions and cations, so let's take a look. A double replacement reaction will occur if one or both of the products is an insoluble precipitate, a liquid, or a gas. To check each of the products you use the solubility rules. You check each rule in order for each of the products. Here is a link to the rules I found online:

http://www.dorettaagostine.com/Solubility%20Rules.pdf 

Look at the double replacement reaction chart. The rules are followed in order. Let's look at $\ce{KNO3}$ first. The first rule states that alkali metal salts are soluble. Potassium is an alkali metal, therefore potassium nitrate is soluble. The second product is $\ce{Pb(OH)2}$. Looking through the rules, none of them apply until the 5th rule which states that hydroxides are insoluble. Therefore lead hydroxide is insoluble. 

As you can see, 1 of the products is insoluble, therefore the double replacement reaction will occur. I hope that helps! If you have anymore questions or clarifications just ask.

EDIT: How to find the net ionic equation.

First reduce the equation into it's respective ions.
$$\ce{2KOH(aq) + Pb(NO3)2(aq) -&gt; 2KNO3(aq) + Pb(OH)2(s)}$$
...becomes...
$$\ce{2K+(aq) + 2OH-(aq) + Pb^2+(aq) + 2NO3-(aq) -&gt; 2K+(aq) + 2NO3-(aq) + Pb(OH)2(s)}$$
However, keep in mind that $\ce{Pb(OH)2}$ does not break down into ions because it is insoluble, as we discovered earlier. Therefore it stays in it's full formula. At this point you do the "cancelling out" by removing duplicate ions on each side of the equation. The cancelled ions are also known as spectator ions because they do not actively participate in the synthesis of the product.
$$\ce{2OH-(aq) + Pb^2+(aq) -&gt; Pb(OH)2(s)}$$
The potassium and nitrate ions are removed because they appear on each side of the equation in the same quantity. These cancel out leaving the net ionic equation show above.

The net ionic equation will show the creation of the insoluble, liquid, or gas precipitate that forms as a result of the reaction. I hope that helped!