Sounds like you have a leak in the sink trap or you are siphoning water out of the trap at the sink which compromises the water seal and allows sewer gases into the house. Do check your vent and check your trap for leaks

To me, it kind of depends on how deep the new sink is. If it can easily hold an entire laundry load of water without overflowing (in case of plug-up), then go ahead and get an undercounter sink. But if an undercounter sink cannot hold that much, I would worry about being outside, or in bed, or away from home while a load is washing. Having the ability to drop a load in on the way to work and take care of it when I get get home is one of the perks of having a home laundry.

You could adapt an outdoor shower faucet/ exposed shower faucet. Use a shorter standpipe and connect a hand held shower to it. https://www.homedepot.com/p/2-Handle-1-Spray-Outdoor-Exposed-Shower-Faucet-in-Chrome-Plated-Brass-Valve-Included-67252/301893008 Another possibility https://www.thebathoutlet.com/Shower-Faucets/product-search/10487?gclid=EAIaIQobChMIoeSF1Krs4QIVA7nACh0cowQ9EAQYASABEgL-svD_BwE

probably the easiest thing to do is bump in the wall behind the sink to about 6-8" above the height of the cabinet. Then centre the cabinet on the bump. This will give you a shelf behind the sink; bring the sink forward so you're not squeezed between wall and washer; and make the sink feel equal in visual weight to the other laundry machines.