When I first read your post and saw the print I had the solution worked out in under a minute however it has taken me the better part of two days to figure out how to explain the solution in words without creating what seems like a long list of contradictions in terms. I hope I have succeeded, if not then we will just have additional questions to resolve. First of all I would like to point out that in your post you repeatedly stated that the fixtures dumped into vents. Technically speaking that is not true. A vent is a dry pipe that is intended solely for the transfer of air or sewer gasses. Fixtures must discharge into a drain or waste line and under no circumstances may a vent line run horizontal below a point 6" above the flood level rim of the highest fixture served by that vent. The prohibition against horizontal vents below the flood level rim of the fixtures should be easy to understand. If we have a dry pipe running horizontal and connected to a drain line there is a strong likelihood that solid particulates from the drain line could back flow into the vent line and in time they would build up and block the vent. The solution to your bathroom is to begin at a point in the wall directly behind your lavatory sink. You will install a sanitary Tee and stub out a waste arm to the sink trap. (Careful here, you must use a Tee because a Wye&1/8bend or combo is prohibited) From the top of the tee you will install a vertical line up into the attic space and through the roof. This section of line is properly defined as a "Vent" From the bottom of the tee you will drop a vertical line down below the floor, then turn horizontal and run straightforward to the 3" drain line. The connection at the drain line must be maid by means of a Wye&1/8bend or a Combo (A tee is prohibited here). This line is properly defined as a "Combined waste & vent line". This may seem like silly semantics but I can assure you that the code restrictions for vents or combined waste & vents are much different. By example, a vent may not run horizontal for the reason explained above but as you can see the combined waste & vent does run horizontal. The difference here is that even though the line is primarily serving as a vent for the 3" branch drain line, by configuring it as a combined waste & vent from the sink the line is repeatedly washed out by the discharge from the sink so we need not worry about solid particulates from the 3" line back flowing into the vent line and causing a clog. All the other fixtures in your new bathroom are within the maximum length requirements from the trap weir to the vent opening so once you have the line from the lavatory installed you will not need any additional venting in this bathroom. Figuring out the venting was easy but figuring out the required size of the line or the manner of termination is a bit more complex because the International Residential Code is radically different from the Uniform Plumbing Code. Under the International Residential Code all structures are required to have one "Main Vent" and when additional auxiliary vents are required we may reduce the size of the vent line to the diameter of the line they serve. Understanding that we are venting a 3" branch drain it then stands that the auxiliary vent must be 1-1/2" except, if the total developed length of the vent line exceeds 40Â we must increase the diameter by one nominal trade size. The IRC gives us a number of optional methods for terminating a vent. The preferred method is always to terminate through the roof however the IRC will permit terminating the vent in the attic space (never a good idea), through the wall subject to some rather strict limitations, run horizontally and connect with another vent before going through the roof or we may use an unlimited number of AAVÂs (air admittance valves) however AAVÂs should only be used as a last resort. The table of fixture arm size, maximum length from trap weir to vent opening and required vent size listed in the previous post is based upon the Uniform Plumbing Code however it is not correct. The UPC does not determine the vent size from the drain line size. Under the UPC a vent line size is determined by the DFU (drainage fixture unit) load on the drain line drain line it serves. Consulting UPC table T7-3 we find the DFU load is: LavatoryÂÂÂ..Â.1dfu ShowerÂÂÂÂ..Â2dfu Watercloset (1.6gpf).3dfu Total ÂÂÂÂÂ..6dfu UPC table T7-5 allows up to 8 dfuÂs on a 1-1/2" line, except if the drain line serves a watercloset the minimum is 2" therefore under the UPC to vent your bathroom group we would be required to run a 2" line as described above. The UPC does not require a main vent but the combined aggregate cross sectional area of all vents must be equal to or greater than the cross sectional area of the house main drain. Under the UPC all vents MUST terminate through the roof. Vents may be combined in the attic space before passing through the roof providing that when two or more lines are combined the resultant line size is increased to maintain the total cross sectional area. In addition, under the UPC if any vent line is offset horizontally for a distance greater than 3Â horizontal the size of the entire vent line must be increased by one nominal trade size. Here is a hint. In plumbing any line that runs at an angle of 45deg or greater above horizontal is said to be a vertical line, while all lines that rise at an angle of less than 45degrees are said to be horizontal lines. You may offset to the backside of the roof peak, as you desire without increasing the line size if you start right at the attic floor and make the offset rise at a 45degree angle.

I'm not sure what you mean by running a vent 'off to one side'. If you are wondering about the horizontal run, it must be at least 6" above the flood level of the highest fixture serviced. Though you don't give the measurement between the lavatory vent connection and the shower drain connection, from your drawing it's apparent that you do need the additional vent.

The installation of a new vented fan isn't difficult if you hire a professional to do it. If it is 'very old' as you say, then I would recommend replacing the entire assembly: fan, duct & duct cover. Newer fans have 'dampers' that prevent cold air from coming back in & a newer model would vent your bathroom properly & safely as well.

The main reason is laziness and cost. What they should have done is installed an insulated 4 inch flexible duct to a capped roof vent. It needs to be insulated so the warm air does not condense in the winter and drip into the fan. Your old fan probably got rusty because there was no roof cap and the duct was not insulated.