Theres one thing I need to clear up. I forgot you had a civic and The a/c test I mention above does not pertain to civics because civics do not have a full radiator and full condensor. But you definitely got it working with the old radiator so everything looks good to me. The little thing you're talking about is probably some regulator of something. I don't quite know what it is casue I've only seen it on some civics. I have a 95 coupe and I don't have it but my brother's 92 hatch has it. I'd have to look into this but never has had any problems for my bro so unless it's been modified, don't worry about it for now.
About the car running hotter than normal. I think is time to check the whole system and replace the thermostat, radiator cap and inspect the radiator for cracks, and any clogging.