The entry for adiuvare in L&S indicates that the verb is indeed used with the accusative.
The only exception mentioned is the possibility of saying laborem tibi adiuvo, "I help you with work", where the person to benefit is in the dative and the thing they receive help in is in the accusative.
This seems to be rare, so as a general rule the ...
Indeed, it is accusative plural! If you have an edition with long vowels marked, you'll see that this is fortīs, as opposed to nominative or genitive singular fortis.
Ad-, though, isn't the key here. In almost all cases, if a Latin verb takes a subject and an object, the object goes in the accusative case. There are a handful of exceptions that take genitive,...
It is the second option with arbitrata.
For the purposes of agreement, you can think of the participle as an adjective, so that Syra arbitrata est and Syra Romana est have exactly the same form.
The gender agreement holds for all subjects of all numbers in the same way.
For example, uxores arbitratae sunt and mariti arbitrati sunt.
And there is no difference ...
This is what Weiss 2020 writes on cognomina in -a (in III. *eh2-stem suffixes):
Aemilius Alba, from albus 'white'
Calua, from caluus 'bald'
C. Mucius Scaeuola, from scaeua 'left-handed person'
Weiss also mentions Klingenschmitt 1992 - I'm going to re-read this and add more later - who explains these cognomina as collectives with the meaning 'family of X' ...