Those scooter engine aren't designed to run at high power settings for hours on end; the engine is likely to wear out quite fast.
Even with a professionally developed gen head, engine-power-in =/= electrical-power-out; the ratio is something like 65-75% in the better cases. There are a number of reasons for this, but, what it realistically means is you're only likely to get maybe 350 watts or so, assuming 500W in. You're not going to power your house off this thing.
2 stroke scooter engines (or, really any small 2 stroke) want to turn REALLY FAST (like 8k to 12k rpm) to generate useful amounts of power. Leaving aside the LOUD factor, you're either going to need to build speed reduction and speed REGULATION systems (the Briggs and equivalent engines have a governor built in, 2 strokes DON'T), or do your 60Hz waveform electronically. The speed reducer or the electronics are going to cut your usable power back even farther. . . . so now you're maybe going to see 250-300W peak. It's going to struggle just with the fridge and a couple CFL bulbs.
So: Couple the foregoing with the fact that 2-stroke mix goes bad even faster than ordinary gasoline, and you can see how a mini-generator maybe isn't the best option for emergency stand-by power.
BUT-- so much for the caveats. You're looking for a project!
Now Honda (amongst others) made a 2 stroke powered genset of about 500-1000W; There were a bunch of electronics involved, and effectively the units were a computer UPS with an engine and generator in place of a battery bank, and a MUCH bigger fan and heat sink. The engine only ran as fast as needed to balance the load, so unless the unit was highly stressed, (trying to start the fridge compressor, maybe?) it just putted away quietly.
Which then points the way forward:
Source a cheap car alternator, a clapped out UPS that still has batteries (that maybe don't hold much charge, but still register 12V; you'll need the batteries to keep the voltage regulator on the alternator honest and your electronics alive!), some pulleys and belts, a stout block of wood (not particle board!) and some rubber to isolate the various vibrations, and you can probably take it from there with a hand drill, a soldering iron, some heavy gauge wire and a couple of end-wrenches. Use nylock nuts! (lock washers don't work well on rubber isolated applications.) A deluxe version would have a voltmeter and ammeter on the line to the batteries, so you could futz with the throttle setting to JUST balance the load. . . . but you don't HAVE to have that.
A 2-1 reduction will probably do; alternators like to turn somewhere around 6k-9k RPM or so; some of the Japanese units can go 14-15k before they grenade. If the alternator is still in the donor car, you can calculate the max safe speed by measuring the diameters of the crank pulley and the alternator pulley, and looking up the engine red-line. . . . . like maybe on the tach?
The result should be able to power your laptop, the broadband box and a couple of lights indefinitely. Efficient? NO. But it WILL work. As a bonus, you could rig it as a heavy-duty portable battery charger/jump-box.