Is it possible to realistically use Lydia for the entire game

Any follower that is skilled at light armor and sneak (and optionally, archery too) should be fine. Followers should preferably be non-mage, as casting spells will attract enemies and blow your cover. Complete list of followers and their descriptions here: http://uesp.net/wiki/Skyrim:Followers

Followers that are good at sneak, light armor and archery are: Aela the Huntress, Cicero and Dark Brotherhood Initiate.

Tips to make followers extra-sneaky:

• Have them wear light armor only, as heavy armor is noisier and makes them more likely to be detected.
• Give him/her an item that has the Muffle (wearer is muffled and moves silently) effect (usually a boot or a shoe; and usually light armor too).
• Have you follower wear apparel that increases sneak skill (Fortify Sneak).

Also, note that (from the UESP wiki's "Skyim Followers - Follower Stats" article):

A follower levels up according to the PC's level up to the follower's maximum level (which varies for each). With few exceptions, each follower has a maximum level, and a specific set of primary skills, which are dictated by their class.

Prior to patch 1.6, a follower's skills, health, and other stats were set when they were first spawned (which is typically when you enter their starting cell, become a Thane, and so on) and would not increase as their level increased. Prior to the fixing of this failure of stats to update with patch 1.6, players could update a non-essential follower's stats so they would increase based on their current level by using the Wabbajack to polymorph them into another creature, after which their stats would update upon recovering their original form. Players of the PC version could also use certain console commands to fix any follower's inability to update their stats with each level.

So in most cases, leveling up the player character will also make your follower level up, increasing the follower's sneak skill (the maximum and by how much, will depend on the specific follower and their NPC class).

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.