Is my beam support wall supposed to touch the beam exactly, or is a small gap okay?

Your neighbor is right, the wall has to actually be touching the beam to carry anything. Wood doesn't really 'settle into' a gap like that. The beam might sag a hair over the years and pick up the load unevenly, but you can't count on it, so fill the gap now while everything's open. I ran into the same thing with a support wall I framed under a girder, and the fix took about an hour.

For the quarter inch end, use pairs of wood shims driven in from opposite sides so they taper against each other and lock tight. The important part is snug, not hammered to death. If you really crank on a shim under a beam like that you can actually lift the floor above a little, and then you've created a hump you'll chase forever. Snug means it resists a firm push with the hammer, that's it. The camber is normal, expecially on older beams, and you don't fight it. The middle is alredy touching, so you just fill whatever gap exists at each spot along the plate. Check it in a few places with a straightedge or just slide a shim along the gap and mark where it stops grabbing. If there's a bigger low spot than a shim can handle cleanly, that's where a bag of non-shrink grout earns its money. Trowel it under the plate, let it cure, and you've got solid bearing the full length. One more thing while it's still open: run a level across your top plate before you button it up. If the wall itself is leaning beacuse of that unlevel slab, fix that first, then shim the beam gap. Way easier to correct a stud or two now than after drywall. Sounds like your framing itself is solid, this is purely a fit-up issue and totally normal on a first wall.

One thing I'd double-check before the shims even go in is what's under the wall, since a bearing wall is only as good as the slab it sits on. When I did mine I found the concrete varied a lot in thickness along the run, and my inspector made me confirm it was thick enough for the load before he'd sign off. Did you pour that slab with a bearing wall in mind, or is it just a regular floor pour? If it's the latter it might be worth a call to whoever did your permit paperwork, because shimming the top while the bottom slowly cracks is a lousy trade.

One way I check if a bearing wall is actually doing its job is the level test after shimming. Put a small level on the floor up above the wall and watch the bubble when you snug the shims, it should tilt just a hair toward the wall. I did this on my last basement job and I could see the load transfer immediately, no guesswork involved. If the bubble doesn't move at all the shims are decoration. Also, sneak a second look at those shims a month later, sometimes they creep back out a little once the wall takes weight, and a tap with the hammer tightens them right up.

I'd push back on the grout suggestion. Grout bonds to concrete, not to a wood plate, so what you'd end up with is a mortar bed that's stuck to the slab but not to the top plate, and it'll crack loose the first time the wood moves with the seasons. Around here the usual fix for a gap bigger than shims like is a cutoff of PT 2x4 or two stacked, planed or sanded down to the exact height, sitting on leveling shims if needed. Wood to wood, that bears properly. If the gap is really that big along most of the run, the wall might honestly be better cut and rebuilt to the right length.

Update: the level test worked. Snugged the shims in stages and coud actually see the floor above tilt toward the wall, so I know it's picking up load now. Still stuck on the beam's camber though. It's low at both ends and crowns in the middle. Do I shim the ends and leave the crown alone, or pack it along the whole run?