He is probably juggling a family life, work and other personal commitments along with his online life and has to schedule chatting with you based on the other not-so-secret parts of his life.
His responses may be erratic because he tries to fit in a quick email amidst the other busy aspects of his day or as mentioned above, he might be very regular and follow a strict pattern so keep an eye out.
To search for their profile picture, save their social media profile pictures (and any other likely candidates for dating profile pictures) and drag them one by one into the google images search bar.
If your spouse is using one of these photos for an online dating profile, you may see it show up iin the search results.
The first step is to be aware of common dating sites and apps (many of which are intended to look inconspicuous) so that if you stumble upon them on a shared devices's apps or browsing history, you’ll recognize them.
If you suspect your signifigant other might be using a dating site or app, there are a few ways to confirm your suspicions, including through starting your own infidelity investigation.
You'll need to check their phone for common dating apps, conduct a Google search using personal details like their email address and, if necesary, create test accounts on popular apps and sites to see if your signifigant other is returned as a potential match.
Ultimately, if you’re worried that your significant other is on a dating site, there are many options for sleuthing (ranging from a quick scan through emails and browser history to more intensive—and expensive—actions).
If nothing else works (or you don't have the time to conduct an exhaustive search that may or may not be conclusive), consider hiring a private invesitgator.