Just a quick post about this, it's late so I can't go into great detail, I might update it in the morning.
Should I be leveling new frames, weapons, companions, etc asap? I think I'm MR 5 as of last night. As soon as something is max level I switch it out to level something else. Should I be focused on leveling everything that I have asap? If not, when is the right time to worry about my MR and leveling?
Short answer is
yes, build and level all the things. MR is a time gate that will get you access to some different weapons. Most importantly it exposes you to new weapons, playstyles and teaches you the niches that certain frames, weapons and mods will utilise. heh, diversity.
Second question is each mission has a recommended level. Is that frame level? Weapon level? Does my frame, weapon, companion etc have an aggregate level? Just seems like some missions have been complete ass unless I'm using the ez mode frames and weapons.
Recommended level for missions is not actually what it says, that level number is the level of the enemy types you will encounter. Bringing a rank 2 Lato (sidearm) into a level 40 map isn't going to be fun for you. For things like survival missions, that number is the starting level of enemies you fight, and things will scale up from there. Mission type is far more important to weapon selection and warframe type.
Edit: I have one more question. Are there frames that are considered much better than others? When I looked for a "best" frame or tier list the community seemed all high and mighty about "its all about preference" but then people bitch and moan on Reddit about the Limbo prime access and how it's not a noob friendly frame.
Yes, warframes have niche roles. Some have more utility than others, some have more utility outside of their role than others. Not all warframes are equal, but weapons are the great equalizer in this situation. The list I posted earlier is a great set of 5 frames that you can use for pretty much the life of the game. I failed to mention that Ember doesn't scale with damage very well, and hits a wall that a frame like Saryn does not, I'm actually experimenting with Saryn now, and while Saryn scales into high level enemies, Ember outperforms her at anything before level 50 (mob level).
To put things in perspective if you've done it... I regularly solo 20-30 minute Orokin Derelict Survival with Ember or Nova. At 20 minutes Ember starts to really fall off, and Nova is just hitting her stride and will scale with your weapons past that 30 minute mark. While 20-30 minutes solo T4 survivals aren't all that much of a big deal, it's pretty efficient to just run to 20 and restart if you're farming warframe components.
Side note about frames.. and why I believe getting all the things is important.. trying to get the Kavat scans with any other frame other than Vauban can be painful. It becomes trivial with Vauban (his Bastille suspends all hostiles in range, making defending the Kavats from the enemies on the map, and actually scanning the jumpy little bastards easy). Vauban was also super useful on Infested maps along time ago, that I believe now that a bunch of Infested are ranged, Vauban was not as easy mode as it once was. I also bet that there's another way to scan Kavats just as easy as Vauban that I haven't thought of, or read about.