I can't really comment too much on the Abyssea/Low-man situations since it's been a very long time since I did them on Summoner, but as far as soloing, there are some factors you'll want to take into account.
For example, do the mobs you're fighting use primarily physical or magical damage? If magical, are they all based on the same element or varied? Do they have status ailments that a certain avatar, based on its element, might be able to resist? Can the monster be stunned? Does it take more physical or magical damage? etc. Depending on these things, the avatar you use and the method you go about attacking them will be varied quite a bit.
For most of the solo fights you will do early on with Summoner (starts at 65 with Fenrir but is expanded to most all the other avatars at 70), it'll be a matter of sending your avatar at the monster, using their strongest physical bloodpact (Eclipse Bite, Predator Claws, etc) and then, depending on how strong the monster is, you'll want to run away from the fight until you can no longer see your avatar's HP bar. Once you're that far away, wait until the avatar dies, then resummon it and let it auto-attack the monster. In a solo situation, you will not need to use Assault after the first one since the avatar will auto-attack the monster to protect you. From that point on, just rinse and repeat until the monster is dead. We don't necessarily kill very quickly, but it's smooth and generally safe.
Later on, depending on the mob, you might find yourself using the merit Bloodpacts because they're magic damage over the physical damage from the lv. 70's.
The MP issue later on won't be a problem due to gear you'll come into that lets you cut down on the perpetuation cost drastically, letting you keep all your avatars out for free the same way you can already for Carbuncle.
As for endgame, that depends on what you're doing. In Voidwatch we're called upon primarily for just our unique line of procs, which include the lv. 60 tier IV spells from the Celestial Avatars, Lunar Bay from Fenrir, Night Terror from Diabolos, and Holy Mist from Carbuncle. Apart from that we're just there for damage and cures like usual. In Legion, we're primarily there for Perfect Defense, Alexander's 2-hr ability, and Ramuh's Shock Squall bloodpact. There are some specific mobs in there, such as the adamantoise mobs, which we're useful for since Shiva's Heavenly Strike merit bloodpact can do decent damage to them. In Neo-Limbus, four Smn, a Brd, and a Cor can take down Arch Ultima with our merit bloodpacts, and with a Paladin and White Mage to tank him, can kill Arch Omega due to having both strong physical bloodpacts and strong magical bloodpacts.
I hope this helped you a bit.
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