Missions & Complexes

Since your Eveboad has a password I am going to say Vargur. Others will say Golem, or Mach or RS, but I like the Vargur. Its personal preference really.

I fly Rattlesnake.

tengu is a good fit given SP. I'd just go with a small shield booster as it is more than enough for lv 4s. and you probably don't need rigors and 2 target painters.

also probably worth it to get a raven so you have some alternate damage types, and with some more skills probably more dps. and since you have the isk to spend get a Raven Navy Issue.

Currently flying a Kronos (for the hell of it) with some reasonable results, but by no means the most efficient ship I've used.

I miss my rattlesnake with sentries - pretty much unstoppable in l4's if you want to semi-afk burn through NPCs. Mach is a bit more "Hands on" though more fun if you have the time / attention and really, really enjoy shooting at red crosses.

To be honest, I don't remember why I sold it... I suspect it was so sexually arousing that it was becoming unhealthly for me to fly.

I would suggest a navy scorpion:

high: 6 cruise + tractorbeam
mid: large shieldbooster + sba, 4 hardener, 2 painter or 1 painter + ab
low: 4 bcs + powerdiagnostic system
rig: 2 rigor, 1 flare

You currently have still low skills for a battleship. The sni has a strong tank, so that you can make mistakes and still survive.
The next step could be a cnr.
After nearly maxing out the cnr in the next 4 - 5 month, i would start crosstraining in a rattlesnake or typhoon. After that i would start with a basic gunnery training in large guns and also train amarr bs. That means in about 9 month you should be able to use all battleships for lvl 4 missions.

Yep. With these skills either Rattlesnake or Dominix for L4. Given that you have 3 bil to spend I would go with Rattle. Tengu is nice ofc, but it's far too "loot pinjata" given it's low EHP if caught with hardeners offline by a instalock nado at station undock or gate. Besides it's not as effective at L4 missions as Rattle or Dominix.

Do not overpimp it. You know silly useless stuff like C-Type invuls and such. Better to have a little reserve than spending it all on one ship, dying and then being broke.

Fit, well ... it can be a bit tight at 13 mil SP.
Cruise launchers, Drone range
Tank, propulsion and perhaps few omnis or drone nav computers
damage control + damage mods.
Rigs - whatever you can pull off with your skills. Ideally you would want T2 warp speed and their either Rigors or mix of drone and rigor, but it's gonna hit CPU pretty hard.

Dont forget hardwires. Spend about half a bil of your budget on reasonable selection of these. Stick to 5% ones.

For inspiration - do not just copy paste it, its not a solo L4 fit (flies with logi support). And it will not fit with your skills anyway. Drop the Ogres and get few spare flights of lights instead for solo stuff to kill the frigs in a hurry if you need it. Or carry a second Gecko so you can switch em out if one gets damaged.
[Rattlesnake, L4_wingman]
Ballistic Control System II
Ballistic Control System II
Ballistic Control System II
Drone Damage Amplifier II
Drone Damage Amplifier II
Drone Damage Amplifier II

Drone Navigation Computer II
Drone Navigation Computer II
100MN Microwarpdrive II
EM Ward Field II
Adaptive Invulnerability Field II
Drone Navigation Computer II
Drone Navigation Computer II

Cruise Missile Launcher II, Caldari Navy Inferno Cruise Missile
Cruise Missile Launcher II, Caldari Navy Inferno Cruise Missile
Cruise Missile Launcher II, Caldari Navy Inferno Cruise Missile
Cruise Missile Launcher II, Caldari Navy Inferno Cruise Missile
Cruise Missile Launcher II, Caldari Navy Inferno Cruise Missile
Drone Link Augmentor II

Large Hyperspatial Velocity Optimizer II
Large Warhead Rigor Catalyst I
Large Warhead Rigor Catalyst II

Bouncer II x2
Ogre II x2
Gecko x1

Edit: Do not be afraid of injected setups. Ancillary shield booster can give you very strong tank for a little while to either get things under control or kill the points and bailing it. Also - light drones are reasonable for throwing em out for elite frigates to much on while you bail if thing go truly bad. It's not 100% reliable but elite frigs often drop the point on you to go after your light drones if you put them out.