Also it seems that if the lens is not quite focused at infinity (but the subject is at infinity) there would be some tilt of the focused image.
No, there is no such thing as focusing an image, or focusing a lens - although we talk so much as if there was that it sounds odd to say there isn't.
If you have a lens sitting in the light all on its own it forms a three-dimensional image behind itself, called an aerial image. The whole aerial image is sharp - there is no depth of field. The image is reversed - left-right, top-bottom,
and near-far. Things at different distances in front of the lens are imaged at different distances behind it, and the exact distance is given by the lens conjugate equation: 1/f = 1/u + 1/v, where f is focal length, u is the distance to the object in object space and v is the distance to the image in image space. When u is very large - "infinity" as one might say, exaggerating slightly - 1/u is very small and 1/f = 1/v. That is, objects at infinity are imaged at the focal length (or, the definition of focal length is the distance at which objects at infinity are imaged). As u gets smaller, 1/u gets bigger, and to keep 1/f the same, 1/v has to get smaller, so v has to get bigger. So objects closer than infinity are imaged further away from the lens. (This is why diglloyd's claim that his D850 had a flange focal distance that is too
short - "46.4 instead of 46.5" - so he loses infinity focus is 100% solid-gold no-way-out wrong).
When you use a view camera you get an image of things closer (lower u) by moving the lens away from the ground glass (bigger v). You can't do that with a 35mm camera, because it has a fixed lens mount. But v is measured from the rear principal plane, and that can be anywhere - in the lens, behind it or in front - and it can be moved by changing the position of lens elements. That is mostly how a 35mm lens "focuses". (There is another way: if you look again at the lens conjugate equation, you will see that if u gets smaller, so 1/u gets bigger, but v does not change, 1/u + 1/v is bigger, so 1/f is bigger and f is smaller. So when you have moved the real principal plane as far as you can, and you want u to be still smaller, you can move some other lens elements and change f, which is "focus breathing").
If you tilt the lens or the film, that means that part of the sensor/film has v bigger than another part. That means those parts of the film/sensor have sharp images of things at different distances. It does
not mean that one side of the image is "soft" or out of focus - unless you are photographing something two dimensional. That is why you can use a tilt lens to manipulate the depth of field.