First thing you have to look through is the ORS. The section that pertains to firearms and the CHL is 166. Specifically, 166.291 - 166.295 which is the cream of the crop for the CHL. For you specifically, I'm betting these are the ones that will pertain:
166.291 Issuance of concealed handgun license; application; fees; liability.
(h) Has not been convicted of a misdemeanor or found guilty, except for insanity under ORS 161.295, of a misdemeanor within the four years prior to the application
(L) Has not been convicted of an offense involving controlled substances or participated in a court-supervised drug diversion program, except this disability does not operate to exclude a person if:
(A) The person has been convicted only once of violating ORS 475.864 (3) and has not completed a court-supervised drug diversion program under ORS 135.907; or
(B) The person has completed a court-supervised drug diversion program under ORS 135.907 and has not been convicted of violating ORS 475.864 (3)
ORS 475.864 is what relates to marijuana possession, and 135.907 is a diversion for possession of marijuana. If you only have one violation, then you SHOULD be clear to get one there. As far as the DUIIs go, nothing is mentioned in the ORS about it as far as the CHL is concerned. You may want to double check at the Sheriff's department with their CHL unit, but from what I'm reading you're free and clear.
As far as the legalities of openly carrying on your property, I'm not entirely sure. You would have to check with Tigard's ordinances for that one, but it is YOUR PROPERTY. Don't be a test subject for it because of what I've said, but I haven't read any thing in here as far as personal experiences openly carrying on your own property. If they have a law forbidding openly carrying firearms or regarding loaded firearms, that will still apply to with in city limits if they want to push it. I wouldn't except in the comfort of my own home if I didn't have a CHL, but again, check local ordinances. Another good way to check would be to purchase a pistol once your bench probation is up. If you can buy one, then you probably meet the requirements to get a CHL.
Also, good job on staying sober. Being in the US Army, I've seen too many people get DUIIs (DUI, DWI, or what ever else other states call it ) and screw up there lives. Stay safe.