I think your answers are: No(or maybe), yes(under certain conditions) and maybe.
FAR 43.3(g) states:
"The holder of a pilot certificate....may perform preventive maintenance on any aircraft owned or operated by that pilot which is NOT used under Part 121, 127, 129 or 135."
Now before you go running off, 43.3(h) has an exception for operating rotorcraft in a remote area (and since you listed transmission, I'm thinking you're flying rotor stuff). So if your operation has this exemption, well carry a small toolbox! ;-)
For normal stuff (i.e. you are not exempted) then we go down your list:
Hydraulic fluid - listed as preventative maintenance under the Appendix to Part 43 (bullet # 8). Ergo, 121/135 pilots can NOT do this unless otherwise exempt.
Engine Oil - this is NOT listed anywhere BUT Part 135 DOM's (Directors of Maintenance) have to keep a very strict engine log. Generally, one sign of wear is oil consumption and most operators require a "log" for the pilot to note oil added. Ergo, a pilot should be able to add oil or have a line boy add oil, but you "should" have to note this or log this and hand in the log to the maintenance dept. Your Operations Manual should cover this operation.
Transmission Oil - may be in the Appendix under #6 and concerns the lubrication of parts that only requires the removal of a cover plate. I am not helicopter rated and have no idea if this is considered a special lubricant or is just considered as another engine oil.