No A2A
The F-105 radar actually had an air-to-air mode, primarily radar ranging for lead computations with some search capability (With my eyeballs I could generally see much further) and it was unlikely you would need exacting lead computations when any likely opponent could easily turn better. The only USAF fighter it could out-turn was the F-101 and to air refuel smoothly on a 135 at 310 KCAS (P&D or B/R), the leading edge flaps had to be extended.
In the late 70s the air-to-air mode was not maintained, but to my surprise in 1979 I got in a tail number one day where it actually worked. It also had a small, almost unreadable head-up radar repeater in the combiner, really in fact not much good for radar bombing or a2a search anyway.
I think it's ironic that the Navy makes a point of calling the F-8 the last gunfighter....if it was, it was in name only.
(The F-105 was generally flown by the USAF's best pilots early in the game (a lot of ex-Hun guys), which probably accounted for most of the successes they had, but was opened later to lesser experienced Majors & above {not even necessarily fighter types} wanting to get their career ticket punched. At this time many of the 105 guys had opted for additional tours in the F-4.)
F-105, even in 1979, would have been tasked many days/weeks after D+1. The old girl was not as quick as she used to be. So F-35 comparison is really moot.
(BTW I don't consider myself a Thud driver (they all wear the 100 over the North patches), but I was a dilettante F-105 pilot and enjoyed the hell out of it.