Most Tennessee jurisidctions appear to use the International Plumbing/Resdiential Codes
so don't be presume there isn't a plumbing code requirement were you live.
Glennjannie is correct in that most codes (like the IPC/IRC) require a minimum of a 2" floor drain and local variations may require large waste receptors. He is also correct that sinks and washers cannot drain directly into a floor waste receptor.
However, there are a myriad number of ways to create a trap primer by having the standpipe from a washer or sink prime the trap.
But if you want a long term "worry free" trap primer, you might consider one that work automatically by intermittantly opening a valve on a water supply line to fill the trap on a timed basis. Works kind of similar to an automatic fill valve on a boiler. This way you won't have to rely on anyone using a drain or fixture to keep the trap filled. Here is an example:
You should probably contact your municipal plumbing code enforcement office to get their feedback as to what is acceptable.