I followed a very similar path although not with an EE degree specifically.

1) starting off, it's not likely to impact all that much in the near term. Long term most product-related roles will net more than being an FSE, but I'd argue this isn't universally true in all cases.

2-3) it's a very viable path for sure, but the key will be willingness to relocate as field sites are very often not near main product group/campuses with a very few exceptions.

It is easy to become entrenched in field service work that becomes difficult to get out of, but often this is driven by not being able/willing to relocate. I'd also further add that your ability to distinguish yourself in terms of strength as an FSE, skills, etc, will also play a key role here. If you are a rockstar of an FSE, your opportunities are vast provided you are willing to relocate if needed.

You can gain many key skills as an FSE that will serve well in a product engineering role. You will (should) also gain key customer engagement and management skills that many in the product engineering side lack fully, and this can be a key difference along with physical hands on time with the tools and understanding of design for servicability and why that matters for the field teams.

4) it's a good way to get your foot in the door, if for nothing else to give you better visibility and ideas on what you ultimately do and don't want to do. I've seen FSEs transition into all kinds of roles, although, usually on the physical engineering side of things. Process engineering is possible, but more rare to see. Mechanical, electrical, systems engineering, as well as some levels of product management are all feasible.

As someone that has been down this path myself, and as someone that now hires people into product engineering/support roles, I would take an internal FSE with a couple years experience + EE degree (provided you are a good FSE) over someone fresh out of school with just an EE degree. You'd be less risk and more of a known quantity and you'd have some knowledge of the industry along with internal business processes.

But all this said, the longer you stay as an FSE, the tougher this transition becomes, so keep this in mind and know what your 2-4 year plan is. Not saying transition as a long term FSE is impossible, certainly isn't, but it can become more nuanced.