The best oil for your 360 is Rotella in the blue bottle. Ill have to walk out to the garage to find out weight, but the blue bottle only comes in one weight. If its cold, yes mine also clunks into first. It does sound a bit like there may be a problem but it should smooth out a little as it warms up. Im on my 4th season with this 360.
If you ever pull the right cover off its a great idea to replace the clutch springs with heavy duty ones. And new clutch disks are not a bad idea. Its a pretty easy job and they are pretty inexpensive. And, while the cover is open you could add the updated Honda oil pickup. Another easy ob. They are cheap and usually readily available on Ebay.
clutch springs...labeled as 350. However, one of the only parts that are interchangeable w 360.
http://bore-tech.com/product/350-clutch-springs-cs-4
Here is a link to free manuals page available on my website (including the oil pickup update from Honda)
http://www.hondabrat.com/search?updated-max=2013-04-18T03:29:00-07:00&max-results=7&start=15&by-date=false
And I just swapped out my oil pickup this spring...here is a couple posts about that. If you open the oil filter can to clean it out, make sure you have a new o-ring for it. available at your local Honda dealer. Mine old one had swelled up and I couldnt reuse it.
http://www.hondabrat.com/2014/03/do-as-i-saynot-as-i-do.html
http://www.hondabrat.com/2014/03/oil-o-ring-grrr.html
