Perma +20ft movespeed, uncommon item, very good!

Does this also mean they can jump 30ft vertically/diagonally to get on top of tall objects?

This is my question as well.

It doesn't specify long jump or high jump. Which means all of your jumps.

Does the speed of 30 overcome conditions that impose a reduced movement or speed of 0? Does taking stead aim actually impose 0 speed while wearing these boots of 30 speed?

Actually, no.  Jump is limited to your movement, so assuming your walking speed is 30, you can never jump more than 30 ft....UNLESS you dash. https://www.dndbeyond.com/sources/dnd/phb-2024/rules-glossary#LongJump

Isn't everyone's character's Speed now 30 feet (no more 25 feet for halflings, dwarves, gnomes, and such)?  Would these give 30 feet of Speed if something has reduced your Speed?

Sure, why not?  It says "... you can jump up to 30 feet..."  (srsly tho)  In 2024 rules, you're limited horizontally by your STR score (in feet) for Long Jump [so likely 20 feet max] or vertically by 3+ your STR modifier (in feet) for a High Jump [so likely 15 feet max]

I can guarantee my player are going to fight for this item  

I agree

I'm making a Monk / Fighter grappler, these boots will be necessary for the build.  Punch+ grapple, jump 30' straight up release.  They fall prone for 3d6 damage.  I land next to my prone target and follow up with 2 light weapon attacks at advantage. Then use my bonus action attack to grapple again,, so they remain prone the next round.  Repeat.

I am nearly certain that RAI this item is designed to surpass that limitation.

Thanks Daran <3

Perhaps, but RAI isn't RAW. In your game do whatever, but the jump rules are very clear:  "Either way, each foot you jump costs a foot of movement."  and that is used in both high jump and long jump.