When your temp is low 30's or below, the heat pump will not do much, but at 67 degree outside, that is primo operating range, so something is wrong.
Diagosis of sealed systems is not amateur hour. At a minimum, you would need a set of gauges and a good digital thermometer to properly analyze your sub-cooling perfomance. This is the only true diagnostic for your system. But a good tech could interpret some of the more obvious symptoms. Yes, the reversing valve can be the problem. The fact that the small line from outside starts getting cool points to a problem. First thing you need to know, is YOUR system set up to energise the reversing valve for HEAT or COOL ?? ( on your thermostat, there should be a connection to a B terminal, or an O terminal) . A newer thermostat may just have one terminal, and a programmed set up determines how to handle the reversing valve. If you have a newer programmable, or even just digital, stat...get out the install book and check the set-up, after determining which system you have. I noticed you mention a newly installed stat, so that needs to be checked right away.
Reversing valves can get stuck of course, but a more common failure is the coil, which is replaceable.