I found it on the way
Typically I would expect this to end with "on the way home", "on the way to school", "on the way back", or something to that effect. This phrasing to me emphasizes more when or what they were doing when the object was found, as opposed to the object's location.
I found it on X road/street
For both of these, my inclination is to use the article the and replace on with in for the last two examples given.
I found it in the road/street
Using on makes it sound the object found is of comparable size to the road (eg. a car). However, oddly enough "on the street" doesn't sound as strange.
EDIT: forgot to answer the second part of your question:
Yes using it like:
I found it on [street name]
is the appropriate usage. I would also add the often, for certain roads, the "road specification" such as Street, Lane, Way, Court, etc. could be dropped.
I will also add the following suggestions that sound the best to me ( or if no preference):
I found it:
- in the grass
- in the park
- on the sidewalk
- in/on the driveway (same as street)
- on the path
- on/in the walkway (reverse of street)
I would say that sidewalks and paths sound like "surfaces", while grass and parks are "3D". Roads are "what cars use" while walkways are "what people use" (lost items/pens are of closer size to people (?)). Finally streets and driveways are in between all these definitions.
Not sure if that rationalization will help but.... and disclaimer this is coming from someone raised in Cincinnati, Ohio, USA so may not be applicable in different dialects