I have essentially rebuilt the front end of my 2005 Mustang over the course of the last few months. Most recently, I changed out the strut mounts. From the factory, there aren't any any alignment adjustments associated with the strut assemblies. They just bolt up. So, there's shouldn't be any way for the alignment to get out of tolerance. The only thing that is readily adjustable on the car is toe-in. That shouldn't have changed.

Positive caster helps give a car the self-centering action to help it handle better. Well, now I am experiencing the opposite, or sorts. When I turn the wheel more than one rotation, there's a self-turning effect where the wheels want to turn further. Could improper toe-in cause this?

Also, part of the reason I started replacing parts is that the car wandered and tramlined really bad. The car doesn't wander anymore and drives straight, but it still tramlines pretty badly with ruts in the asphalt. I still have stock-sized tires and haven't them at any point while the problems showed. Could this also be reduced with a toe-in adjustment?

 

I guess I don't share your assurance that bolting in new parts will not result in an alignment difference. An easy example is replacing tie rods. Unless the old ones were measured accurately as to effective length before being removed, and the new ones adjusted to exactly the same effective length on install, toe will be different. Also, no 100% guaranty that a strut assembly has exactly the same dimensions/hole locations/angles.

Strut assemblies on some vehicles have camber and castor steps programmed into their tops. For instance, with 4 studs on top, there are 4 different ways it can be inserted into the strut tower. These usually have an arrow on the top of them, or some other notation. Each position has a different castor or camber gradient, need the shop manual for the particular car to figure it out, unless the replacement assembly comes with info.

For fixed-position strut assemblies, the strut to top of steering knuckle bolt(s) can be a cam bolt. A non-cam bolt may have come from the factory. A cam bolt allows tilting of the strut to knuckle connection, before tightening the associated nut to lock it down. I've had these for camber adjustment, supposedly some cars can have castor adjusted that way too, though I have never had one of those (at least that I knew of!).

For most cars, a little bit of toe-in aids directional stability. Zero toe would be the absolute best for gas mileage and tire wear, but starts getting a bit squirrely. Usually, the more toe-out, the worse it gets, to the point that it is really twitchy and a pain to keep going straight.

If you are having problems one steering wheel turn away from the center, that is getting far away from the down-the-road geometry, and into Ackermann's principals. I would think that checking overall alignment to verify the basics would be a good start, before delving into oddities.

All bets are off if you have something moving around badly that shouldn't be, like really bad ball joints.
With McPhearson struts, I don't know how bad the bushings for the lower control arm would have to be before something shows up in driving. I have a lot of experience in how bad the upper control arm bushings can be in a coil spring over lower control arm vehicles (like the Ford Panther chassis), and what weird effects they can have.

Another possibility is bent steering knuckle, bent or bent body attachment of control arm/suspension, etc., but here an alignment test would show something out of wack, that too much adjustment, or an odd type of adjustment, would seem to be needed, therefore, a clue to look for a problem in those areas.