There are tweet ads, yes. They are usually marked as "promoted"
I found it useful to "report as spam" the account they were coming from so at least you only see them once...
I found the new layout to be headache inducing for my english-speaking, left-to right-reading brain so I switched to the Chrome plug-in Silverbird on TVGugs recommendation... and haven't seen the promoted tweets since. Maybe that's because I'm using lists more... but it solved a lot of problems for me.
Your router software should have a log that will show you devices currently active or maybe even historically using your network, just so you can confirm. It may list IP address, or perhaps even device names.