If you didn't outgrow asthma as a child, and still have it as a young adult, does it become COPD when you're older?

Treatment matters. Most asthma can be effectively treated with inhaled medications that would prevent progression to fixed airflow obstruction. Asthma does not lead to emphysema, but if untreated it can lead to scarring of the airways and permanent airflow obstruction.
Yes it can. Uncontrol asthma over long time many lead to COPD in few and not all. If one knows the triggering factors of asthma it needs to be avoided like pets, chemicals, dust exposure etc if one is allergic to it. In addition to regular use of prescribed medication one may consider, regular exercise, getting infection treated right away when it occurs, daily good sleep and regular bowel movement.
Possibly. Asthma is typically reversible, but if you don't treat it optimally, the changes can be permanent. When it is permanent, some people say it "turns to copd.".
Possibily. There have been studies indicating that the constant, low grade inflammation in asthmatic airways can lead to scarring, and potential irreversible loss of lung function, which is copd. This process doesn't happen in everyone, but in some. There is no concensus as to how to prevent the scarring, called remodeling. Keeping the asthma well controlled is the most logical way, but it is not conclusive.