No

Mind blank does not prevent the frightened condition. It prevents the charmed condition, psychic damage, and various divination effects.

The more ambiguous part of the spell states:

The spell even foils wish spells and spells or effects of similar power used to affect the target's mind or to gain information about the target.

This is speaking on the power of the previous protections: immunity to psychic damage, various divination effects, and immunity to the charmed condition. If the spell were intended to work against effects that affect the mind that are not the enumerated protections, those effects would have also been enumerated. Spells only do what they say they do.

Some lower level alternatives:

Calm emotions, dispel evil and good (against certain creature types), hallow (with the courage option), heroes' feast, heroism, and protection from evil and good (against certain creature types)

The Petrified Condition would not be suppressed or removed unless it is magical

Antimagic field has a specific list of effects stating what it is capable of doing - and the one most applicable to this situation would be:

Targeted Effects. Spells and other magical effects, such as magic missile and charm person, that target a creature or an object in the sphere have no effect on that target.

Petrified is neither a Spell effect or a Magic effect, but a condition, similar to being grappled or exhausted.

If you wanted to remove petrification, you need to use a spell that specifically states it is capable of doing so. For example, the spell Greater Restoration:

You can reduce the target's exhaustion level by one, or end one of the following effects on the target:

• One effect that charmed or petrified the target

However, when the effect is applied by a Beholders eye rays, it is considered magical:

Eye Rays. The beholder shoots three of the following magical eye rays at random

Therefore - magical petrification would be unable to affect a target within the sphere, and both the on-going saving throws & petrification condition itself would be supressed if the targets enter it while affected.

Does suppressing petrification stop the on-going petrification effect?

If the petrification is considered magical, it will be suppressed, both by Antimagic field & the Beholders own Anti-magic eye cone.

On a failed save, the creature begins to turn to stone and is restrained. It must repeat the saving throw at the end of its next turn. On a success, the effect ends.

If the effect is suppressed, the target doesn't make a saving throw, but it also states that the effect is only ended on a success. Therefore the effect continues until the target succeeds on a saving throw.