Answers
Machamp and alolan egg will be very (and I cannot stress this enough) ineffective against both of those.
However, they are useful for getting mewtwo and Alolan raichu's counters, aka, Tyrantiar. You should get the machamp's moveset to Counter/dymanic punch (fast move/charge move) if your macham has bullet punch, use the fast tm on it (because it only has 2 fast moves, you'll only need 1 fast tm), if it has Close combat or Heavy slam, use the charge tm's you have to try to get Dynamic punch
Alolan egg you should use the leftover TMs to get bullet seed/solar beam, it will be useful against Rhydon (who is also useful against Alolan Raichu)
Both will be good against Piloswine, a very easy tier 3 raid boss, so you can use that to get more charge/fast tm's and rarecandy, as well as piloswine will be useful when its evolution comes out
If you're in need of some more power, Magikarp's a good pick for a raid boss, getting 400 candy to evolve it might be a pain, but it is worth it.
100% your top priority should be getting that Machamp to have Counter and Dynamic Punch (fast and charge moves respectively).
(you will eventually want a team of 6 such Machamps at minimum).
None of your other pokemon are worth TMs at the moment. Fast TMs will come around easily, so if your Exeggutor has a Grass charged move and Dragon fast move, you can use a fast TM on it to get it to double grass.
If your Machamp already has Dynamic Punch, save that charged TM for when you eventually get a Mewtwo (or if Kyogre/Groudon/Rayquaza return, or new legendaries are released, or you get Tyranitar).
Basically you want Machamp with Counter/Dynamic Punch as he's the strongest fighting type, and fighting is strong against loads of raid bosses (including Tyranitar).
You want Tyranitar with Bite/Crunch as it's currently the strongest Dark moveset in the game, which is really useful against Psychic types (including Mewtwo).
You want Mewtwo as it's Mewtwo, need I say more?
The little one is useful when you get a "Use a super effective charged attack in a gym battle" quest, because there's always a Blissey, Chansey, Snorlax, or something Normal in most gyms, and if they are low on motivation, Machamp will KO them before he gets to use the charged attack, and you won't complete the quest.
What you need to do is to assemble a team to start raiding Machamps so you can get more of them so that you can assemble a team of Machamps to raid Tyranitars. so that you can assemble a team of Tyranitars to raid Mewtwo, so that you can add a psychic Mewtwo to your Machamp team, ....
Seriously, for a Machamp team, a couple of level 27-30 Espeons with Confusion and Future Sight will get you ahead of the clock, and then use either Ho-Oh with Extrasensory and Brave Bird or Lugia with Extrasensory and Sky Attack or Futuresight. Machamp is also weak to Fairies and Flying, so use what you have, and if you need to partner up with someone for a while before you can solo Machamp, then do so. That's what I'd work on if I were in your position - it was what I worked on when I was around your level.
Good luck.
Do not spend that Charged TM. By the time you know what to spend Charged TMs on, you will desperately want more. Likewise for Rare Candy and stardust.
Continue catching and hatching. Eventually you will find more good Machops, Eevees and Abras (both are crucial to Machamp solo teams), Gastlys, Magikarps, perhaps even Dratinis, Larvitars, Ralts or a Chansey.
If you get the following research with a question mark, get them done as the mystery Pokemon are good:
Battle in a Gym 5 times (Machop)
Catch 10 Pokemon (Magikarp)
Evolve a Pokemon (Eevee)
Hatch 5 eggs (Chansey)
Make 3 excellent throws in a row (Larvitar)
Make 3 great throws (Gastly)
Win a level 3 or higher raid (Dratini)
Catch a Dragon-type Pokemon (Dratini)
Also get done any research that rewards TMs or Rare Candy.
If you cannot solo tier 3 raids yet, try soloing tier 1-2 raids. Magikarp, Makuhita, Exeggutor and Kirlia are very useful, while Mawile is only available from raids.