Can I connect a tankless water heater with 3/4" connections to my 1/2" plumbing?
Yes, you can in most cases. Water heaters have 3/4" connections to allow for increased flow and pressure where it's needed and where the pipes are larger, but that doesn't mean your plumbing has to match. The heater will flow at the same rate (more or less) as a straight run of 1/2" pipe in your case. If anything, the heater will perform better at heating rise since flow is restricted by the pipes, extending time in the heater for each water parcel.

To be sure, read the specifications for the product you're interested in.