Seems Legit

Taking a level of rogue (knife master and scout archetypes) and then a level of ranger (trapper and freebooter archetypes) is a legitimate combination. None of the archetype abilities replace other archetype abilities. Obviously, campaigns vary, so confirm the viability and availability with the DM, but this is mechanically sound.

There's Nothing Wrong With It...

But you're trading an extremely valuable class feature in your ranger spells for making traps instead. "But," you say, "I'm not casting spells until 4th level. I don't care." But you might. See, even though you can't cast spells and have no caster level at 1st level, you still have the ranger's spell list. This spell list lets you use magic items with the activation method Spell Trigger (Pathfinder Role-playing Game Core Rulebook 458), so while most 1st-level rangers can use a wand of cure light wounds, for example, you can't, and instead must rely on the Use Magic Device skill or some other method when wands are cheap and simple.

Further, while the game may never publish another trap usable by that archetype again, you can rest assured more ranger spells will be printed--it's a popular class whose origins date back to the hobby's beginning. And you'll never get access to those spells either despite being, at least nominally, a ranger.

Not having spells is huge in Pathfinder because if you're not using magic somehow you're toast, and usually the more you use the better off you are. At low levels, lacking magical access only stings--you won't have a lot of cash anyway and potions are the norm--but at mid- and higher levels when you could've used that wand of freedom of movement (a ranger spell!) to good effect and can't, it hurts a lot more.

Much of the same applies to the favored enemy class feature. Although the extraordinary ability freebooter's bane grants your party a +1 to attack and damage as a move action at 1st level, favored enemy is a +2 to attack and damage and other stuff all the time with no action cost at 1st level, albeit versus a limited class of foes. That last hurts a little, but unless the DM provides no campaign background, what you pick will come up (I mean, seriously, favored enemy (humans) is almost always solid), and that higher bonus is better--especially for a ranger who scouts and sometimes fights alone. Further, feats, magic items, and spells look for the favored enemy class feature on your character sheet, and probably nothing will ever look for freebooter's bane on your character sheet because favored enemy is what the historically popular ranger gets.

By giving up what the core ranger gets you cut yourself off from a lot of what the game expects you to have in exchange for what, in isolation, looks like a pretty good deal. When the character hits the table, though, the theory hits the fan, and you might be stuck really wanting to use that wand of cure light wounds, cast the spell hunter's howl, or wind a horn of antagonism instead of giving everyone a +1 to attack and damage or finding magical traps (which you could've just done as a rogue).

I'm really not trying to sell you on keeping the core class's class features--I seriously have no agenda here--, but the game expects you to have the core class's class features so it keeps churning out stuff that uses them, often to the exclusion of cool stuff found in splatbooks because those just aren't core, hence they appeal to fewer folks. Defying the game's expectations can be costly in the long run, and what's awesome at 1st level isn't always as awesome at 10th level. If you know the game's gonna crash and burn by session 3, this seriously won't matter much (although being able to confidently wield a wand of cure light wounds is awesome all the time), and you can do whatever. But for a long-term campaign, consider carefully before rejecting the core.

Optional Rule

Unfortunately, multiclassing is an entirely optional rule, so there is no way to force your DM to allow it.

If the DM allows for it, the character needs to have a 13 in the relevant stats for both classes. In the case of a Druid/Rogue, this means Wisdom and Dexterity.

As for how the blend will work, there really isn't much combat synergy between the two classes. The biggest issue is that Sneak Attack won't work in Wild Shape. From a utility perspective, there's some hope. Wild Shape may be good for getting into and out of places even the Rogue's stealth can't get one through.

Opinion Aside

Frankly, you're not playing 5E clean if your DM is mired in behaviors learned in previous editions - the changes in 5E are deliberate and mostly well thought-out. You should address this issue with the DM before moving forward. Introducing a new player to a "dirty" version of the game is going be problematic if/when he joins a clean 5E game.