The correct way of using Skyrim DLC’s

The UESP wiki, 'Quest Timing' article has a detailed list of quests (and notes about those quests) that will have a significant impact on other quests and the game, in general. Some of them are due to bugs, and some are because they are designed to be that way.

It also recommends if a quest has to be done early or be postponed before/until another quest has been finished or another game event has happened, to prevent bugs or a negative impact on other quests, or the game.

The whole article is too big to be included here. I'll be adding some of the most notable quest categories and the quests under those categories related to your question:

Quests to Do As Soon As Possible

• Blood on the Ice
  Why: The Necromancer Amulet is a Legendary-quality item very useful for low-level spellcasting characters, especially those interested in Conjuration, and can be sold for a lot of gold for a low-level character. There are numerous glitches associated with this quest, and the best fix is to start the quest as soon as possible and complete it all at once.

• The Forsworn Conspiracy and No One Escapes Cidhna Mine
  Notes: ...if you side with the Stormcloaks in the Civil War and if you complete Forsworn Conspiracy before Compelling Tribute and then wait until after Compelling Tribute to start Cidhna Mine, it will become unavailable permanently. It is also possible to have the same effect result from completing Forsworn Conspiracy, then completing Season Unending before Cidhna Mine, even if you have not joined the Stormcloaks.

Quests to Save for Later

• Dragon Rising
  Why: Completing this quest early ensures a large number of Dragon Souls by exploring, if the player is capable of surviving the dragon's attacks.
  Why not: After completing this quest, Skyrim will be full of dragons. Deferring it until you're in the mood to fight dragons lets you explore the world in relative peace and allows you to learn Words from Dragon Lairs more easily. Note that starting the quest will not cause dragons to appear; only after killing Mirmulnir will dragons start spawning.

• Joining the Stormcloaks or Joining the Legion
  Why not: Joining either the Stormcloaks or the Legion removes access to all the opposing faction's camps and causes "random encounter" fights to occur. Capturing towns has been known to break certain quests.

Quests that Conflict

• Lost to the Ages (Dawnguard DLC) and Mourning Never Comes
  Why: During Lost to the Ages, you are required to trek through Raldbthar, where Alain Dufont lives. If you kill him before starting Mourning Never Comes, you cannot kill Nilsine Shatter-Shield for the bonus reward in that quest.
  Why not: Alain only becomes hostile if you attack him, he won't defend the other bandits in the area. You can kill all the bandits but Alain, and complete Lost to the Ages without affecting Mourning Never Comes. Also, Mourning Never Comes is only available if you choose to join the Dark Brotherhood; if you choose to destroy them, you will never be given it.

So here's what I know you can do (name followed by the quest you should start but not complete):

People

• Any normal follower (or, hey, Aranea Ienith can summon an atronach)
• Delphine + Esbern - Alduin's Wall (Esbern can summon an atronach)
• Brother Verulus - The Taste of Death
• Mercer Frey - Speaking with Silence
• Thonnir - Laid to Rest
• Avulstein Grey-Mane + Geirlund + Vidrald - Missing in Action (the latter two should join you and Avulstein at Northwatch Keep)

Animals

• Meeko (Meeko's Shack) or Stray Dog (random encounter)
• Barbas - A Daedra's Best Friend
• Horse (better yet, Shadowmere)

Summons

• Two summons (with 100 in Conjuration and the Twin Souls perk). Some dead thralls can summon atronaches, so that gives your summons one summon apiece.

which brings the total to you, nine people, two dogs, a horse, two dead thralls and four atronaches. That's 13 permanent and almost everywhere (excluding atronaches and the horse) and 18 potential. This seems pretty hypothetical and I haven't done it myself, so feel free to correct me.

Other

There's also supposedly a glitch (patch status unknown) where you can dismiss your follower, hire someone else, then immediately return to your follower and rehire him or her. I don't know if people use it with the Dark Brotherhood Initiate(s) for a reason or because it only works with them.

Also Odahviing.