FCO Fast Track

I haven't used that at FCO, but you are right to be concerned. That is a tight connection given that you have to switch terminals. I'd definitely buy the Fast Track.

Do you have to go through security again at FCO? That might be the more relevant question
to ask.

Even if you do, 1) you're protected since everything is on one ticket (yes, you might have to
spend an extra night in Rome, but there are worse fates) 2) You'd be 600 euros richer 3) I
admit to being a bit skeptical of the value of something that only costs 7 euros.

But if you don't have to go through security again, then you don't need a Fast Track coupon.
And even if you do, Fast Track is most likely only at the checkpoint(s) entering the gate area,
not at any connection security checkpoint.

I assume I do need to go through security as I am leaving Schengen and going to the US.

I am not certain how well the Fast Track works. I don't have status with United, just one of their CC. From what I read on their website, a status is not required (?). The $8 cost is a great deal if they get my bag for me. I could likely bring my bag with me on the plane (its fairly small) but I'm quite short and dislike having to store it above my head (I know others may help me) and drag it through airports.

If I don't make my flight on United, would I get booked on the next United flight to my city or any Star Alliance flight? There are 2-3 non-United flights that would leave an hour or so later but they have a stop where my current flight will be non-stop.

Your assumption about having to clear security is just that. I don't recall ever having to
go through a second security check on connecting, as long as you stay in the secure
area. But every airport is different. Just saying you should reconfirm. Since T1 and T3
are connected in Rome, my thought is that you freely walk between the two. It's
different with terminals in Heathrow or Munich that are physically separated from each other.

Now, what you may need to go through is a passport check, since you are leaving the
Schengen zone at Rome. That line could be a factor in making your connection, but
the fast track coupon is of no use there.

And, as noted before, even if you do have to clear security again, it won't be in the line
that is for passengers originating in Rome. Most likely that is the only line which has
Fast Track.

I'm not sure what you mean by "The $8 cost is a great deal if they get my bag for me."
Fast Track has nothing to do with getting bags on/off the plane.

If you don't make your flight, United or Lufthansa/Air Dolomiti is obligated to book you
on the next available flight, but there is room for negotiation. No matter what, having
the alternative flight information available will be handy if you need to reschedule, as
you can propose just as much as they can offer.

I was reading this page online - but as I reread, the top is not part of Fast Track - my mistake - and it still wouldn't do that for the extra price.

https://www.adr.it/web/aeroporti-di-roma-en/services-on-reservation

So can I check my bag from Athens and they will transfer to my United flight? Or would I need to pick up from baggage claim and recheck in?

That's good - as long as I make my flight! If I miss it, will they take my bag off the flight?