LA-EA4 adapter has the AF system built-in, so AF will be unchanged and there's no eye-AF. LA-EA5 adapter uses the cameras built-in AF AFAIK s an A7III will give better performance.
older minolta lenses were all screw drive, which means slow AF performance by modern standards. Sony A mount lenses with built-in motors will be faster. APS-C lenses will work, but expect vignetting - I would consider that unacceptable but YMMV.
IF you choose to use older lenses then select carefully. All the Minolta consumer-grade lenses with ilver plastic bodies were useless garbage i.e. 28-80, 28-100. The 28-105 isn't bad for a lens of that era and the 70-210 f4 is good. The various 75-300s vary between poor and usable but not great. Some Sigma 28-200s are usable, but some are very poor. The 28 f2.8 and 24 f2.8, 50 f1.7 are ok for the era. The 50 f1.4 has a lot of coma wide open, giving a beautiful rendering but not much sharpness. Be aware that many of the older and dirt-cheap consumer-grade Sigma and Tamron lenses were pretty poor, even at the time, and are worse now. Check out Dyxum.com for A mount lens reviews if you don't already know it.
I currently use the Sony 50 f1.4, Minolta 70-210 f4, Sigma 12-24 and a Tamron SP90 f2.8 Macro with the LA-EA4 on my A7III.
Lens tech has moved a long way, and modern lenses are both more expensive and better performing than older models.