Children inherit a number of things from their parents. You correctly identify their initial attributes and the skill which their parents have equipped in their last active slot (Chrom is an exception as he will always pass down Aether to his daughters and Rightful King to his sons), but they will also inherit a few other things.
Firstly, only certain characters can have children. They are as follows:
Each of these characters will bear one child. If two of these characters marry, like Chrom and Sumia, then they will bear two children (Chrom's child and Sumia's child). The gender, character, and starting class of each child is fixed with the exception of Avatar who's child will always be the opposite gender of Avatar and the starting base class of the Avatar's mate (with the exception of Lord and Dancer which will default to Tactician).
On the inheritance front, Children will inherit stat modifiers and growth rates from their parents. This will ultimately determine their maximum potential and how quickly they can reach it. In the grand scheme of things, this has little impact, but it certainly doesn't hurt to keep them in consideration when trying to breed a child with specific traits or maximizing stats. Children will also inherit their parent's classes, excluding gender specific classes, like Pegasus Knight, and will directly determine their skill pool.
As a cosmetic effect, children will inherit their hair color from the parent who does not determine the child. That is, Avatar's child will always inherit the hair color of their other parent, never Avatar.
When deciding marriages, it's completely up to you; the only mechanical reason for picking some pairs over others is for selectively breeding children for certain skills or stats. Outside of that, you are perfectly free to decide your own pairings.
As for when to marry, you are free to marry anyone at anytime with the exception of Chrom who's relationship needs to be determined by the end of Chapter 11 for plot reasons. If Chrom is not married, he will marry the character he has the highest support level with (excluding Lissa). If his supports are at the same level, then he will marry the unit which requires the least support points to reach the next rank. If after then there is still a tie, then he will marry the highest female in this list:
Even then, there are some special cases. If he has at least two support points with Oliva, she is unmarried, and he does not have a C support with any other available female, he will marry Olivia. Even further, if he hasn't gained any support points with any females or there are no available females, he will marry a generic maiden.
You will automatically recruit Chrom's child at the end of Chapter 13. Additionally, all children who have married parents will also be unlocked for recruitment. If a child-bearing character isn't married at this point, their child will be unlocked when they are married. If they're never married, their children will never be unlocked.
Besides from slight changes to support conversations with the generic parent, nothing else is inherited from parents.