I don't know your player, but I know something about playing D&D in AL with ADHD. That's my hobby. I also know something about working with young adults with ADHD in structured settings. That's my profession.

We shouldn't be trying to diagnose your player over the internet, so let's stick with the facts as you've laid them out: a player is disruptive, attributes much/most/all of that to ADHD, and you want to try to help them play without being disruptive. None of that is up for debate.

First thing to think about are the specific strengths and weaknesses this brings to your player. (In my experience they're one and the same.) The flip-side of inattention is the ability to quickly task-switch. The flip-side of fidgeting is working tirelessly at a trivial mechanical task. The flip-side of outbursts is spontaneous creativity.

With those in mind it's time to think about how an AL D&D game works, how it interfaces with those, how (much) to highlight areas where D&D plays to a strength, and how to mitigate those situations where D&D plays to a weakness.

• Some games feature long (say, half-hour) scenes and combats, others have multiple storylines/scenes going on at once.
• Some games feature a tactile component, some don't.
• Some tables are happy with first-person improvisational scenes playing out in unexpected ways, others want to stick with third-person or "zoomed-out" scenes.
• You're almost certainly in a public location with other games going on, foot traffic, and lots of noise from other activities.

In no particular order, here are suggestions for how to turn some of those things to your advantage, how to ameliorate others, and how to make use of your player's strengths. All borne from personal experience.

1. Use minis/markers/scrap-paper chits. Even if you'll never sketch out a map, have them out on the table to represent marching order, chatting with NPCs, exploring town. I keep a miniature croupier's stick in my game-bag and this, along with minis on the table that need to be moved every ten minutes, probably occupies ~30% of my fidget-time. This requires some self-control, as they can't be constantly playing with the minis in the middle of the table. But even during social scenes it can help to be all "So, Princess Vespa. At last I have you in my clutches."

2. Ask them to take good notes. It's a second set of ears to catch a detail you made up on the fly. It's a virtually-constant mechanical task if they embrace it. Flipping back through notes to find a piece of information is another side-task they can call on when needed. And you've got copious campaign notes. Where's the downside? This captures ~60% of my fidgetry.

3. Encourage them to sketch/doodle. The entrance to the temple you just painstakingly described. The layout of the buildings you mentioned in this village. Their character's signature spell going off. This occupies the last !10% of my fidgetry.

4. Have them shoulder some game-responsibility. Got 20 goblins' HP to track? This is your player-assistant: they can play their character and keep an ear out for damage done to mooks for you. That sort of regular task-switching actually eases their experience, and it (hopefully) relieves some of your game-running burden. This obviously requires some trust in their honest play, which seems at question. But I'm not sure how to trace that back to ADHD. I think you need to address that separately.
5. Use them as your reference librarian. My AL GM rarely has to look anything up, because I've already got the book open to the page they want. Again, we've ticked up some quick task-switching and tactile engagement, search-and-find simplicity, all hopefully making your life easier. Again, there's some self-control needed here: I always lay it out there for my GM without saying a word, and just close it up and put it away if he doesn't reach for it.

At this point, btw, we should also have solved the "I have to tell him the scene four times" and "he's on the phone" problems. He's likely on the phone because passively listening to something cannot occupy his attention. But "listen and jot the name and one-line description of every NPC, occasionally flipping to a little-used spell's description" may just be the sweet spot for attention and engagement.

6. Choose (your seat) wisely. You're in a super-distracting environment. Pick a corner/side table if you can. It may be tempting to seat this player with their back to the room--resist that. Seat them by the wall, facing everything. IME the visual distractions aren't the problem as much as the conversations at other tables. It's much easier to avert your eyes than it is to avert your ears. This, of course, helps your other players, too. At an AL location with more than two tables or multiple events going on, everyone's working overtime to filter stimuli. This also probably requires someone--likely you--to be good abut getting there on the early side. That's your talk-with-the-tough-player time: ask them to also come on the early side.

7. Schedule breaks. I don't know about your AL, but my sessions often run three-plus hours. I have a player at one site who really cannot sit for more than twenty minutes--your player may be in the same boat. (By the way, it's probably undesirable for all the other players, too.) Announcing and maintaining a practice of drink-break 1.5 hrs in, bio-break 2.5 hrs in can do a lot. If nothing else, it means that during those two times the player's up-and-about they're not missing new activity. But it's not nothing: knowing that there's a scheduled break at minute 90 can make it much easier not to jump out of one's seat during minutes 70-110.

8. Discuss length of scenes, beforehand if possible. "We're heading into a town you've never seen. Do you want to RP the getting-to-know-you-parts for the next half hour, or do you want to approach it some other way?" When your table's decided, you follow that decision. This way your player(s) who might have trouble focusing during those scenes have advance warning and they can bring their attention-management resources to bear.

9. Get in a habit of declaring intent when heading into a scene. A scene where the table's said "we're trying to buy access to the royal ball" does not sound like a good fit for some random-seeming blurted out idea coming from left field. But the scene "let's head to the docks and see what adventure we can roust" is a nice fit for a wild cannon. Get your table into the habit of declaring intent ahead of time, and thus let this (and all) players know when which playstyles are going to be appropriate or not. P.S. knowing your players' intent makes things much easier for you.

10. Give them explicit warnings. During some of your conversations be perfectly frank with them: "I can't have you disrupting play, it's my responsibility to make sure everyone gets a good experience. I will ask you to leave if I need to. But here's the other half of the deal: I'll let you know if that's coming. If it seems to me that you're disturbing someone else I'll tell you with a note. And I'll tell you if it happens again. And then I'll tell you to leave. And I won't hold a grudge." This, of course, is a fair way to treat any player. Discuss it with your site coordinator: they've got your back.

Final thoughts:

• You mention fudging rolls/cheating. It's hard to see how that would stem directly from ADHD, so I didn't really address it. I feel like it needs to be addressed independently, though to the extent that it might stem from boredom/frustration the above might help. This needs more-targeted intervention--the kind you'll find wisdom for if you look at question on cheating.
• You mention misusing class features. I've got a severe ADHD player in one group who regularly claims they've got features and spells five or ten levels out of reach. I simply say "no you don't," hand them the PHB open to the correct page, and skip over their turn in combat until they present an action which is within their character's abilities. (I check back quickly at the end of each other player's turn.) To be clear: this isn't a player trying to cheat, this is a player who--I think--had never read a paragraph of the class description front-to-back before we started.
• This player might be able to play in Adventurers League, they may not. No matter whether you help this player develop these skills or they end up getting perma-banned by the site coordinator, I think you'll end up the better for the effort. Keep up the good work.

UPDATE: Adventurers League changed the "+1 rule" in 2021.

An announcement from Q1 2021 gives the details. Basically, Adventurers League players may now simultaneously use any and all of the following sources when choosing player options like races, classes, subclasses, feats, spells, and backgrounds:

• Player's Handbook
• Volo's Guide to Monsters
• Xanathar's Guide to Everything
• Mordenkainen's Tome of Foes
• Tasha's Cauldron of Everything

Other source books may also be available depending on the specifics of the campaign. The currently active campaigns (as of June 7, 2021) draw from subsets of the following list:

• Sword Coast Adventurer’s Guide
• Elemental Evil Player’s Companion
• Locathah Rising
• The Tortle Package
• Eberron: Rising from the Last War

Note that variants and optional rules must be explicitly allowed, so the variant Human, Half-Elf, and Tiefling races in these books may or may not be available. The same goes for the customized origins in Tasha's. The best thing to do is to find and follow the character creation guidelines in the Adventurers League Player's Guide specific to your campaign, which can be found here.

The revised rules have been implemented in these active guides:

• Historic Campaign Player's Guide (version 1.0)
• Masters Campaign Player's Guide (Dream of the Red Wizards)
• Alternate Campaign Player's Guide (Oracle of War)

However, the active Seasonal Campaign Player's Guide (Plague of Ancients) remains the version from September 15, 2020. It's likely to be updated when the next Seasonal Campaign begins.

The active Guidelines for Using TCoE in Adventurers League Play also predate the change, so must be revised.

Goliaths

Regarding the original question, Volo's Guide to Monsters is on the list of source books available for all Adventurers League campaigns. Because goliaths are described therein, they're allowed in Adventurers League play.

However, under the updated rules, choosing a goliath no longer prevents the use of character options from other allowed source books like Xanathar's Guide to Everything.