How to get Lydia out of the house

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.

The mod you linked to, TCG - Tradable Companion Gear can indeed allow you to remove all of your follower's equipment, even those that are considered part of your follower's "default gear" (doesn't appear in the follower's inventory, and is unremovable in the vanilla game) - the Blades armor, in this case.

TCG is now also part of a bigger follower ovehaul mod, UFO - Ultimate Follower Overhaul. They are made by the same author.

TCG description from its webpage:

• All followers gear can be traded.
• Default hidden bow and arrows are tradable and not respawnable.
• Followers will not revert to default outfits or unequip gear even if dismissed.
• Followers will wear what is placed in their inventory.

With TCG or UFO, just open up the item inventory screen of Lydia and remove her default Blades armor, and then put in the improved Blades armor that you have.