How big of an upgrade do I need?

Everything but the video card.
If your drive and power supply are that old then they had a good life.
This the least I would use.

PCPartPicker part list / Price breakdown by merchant

CPU: Intel - Core i3-8100 3.6GHz Quad-Core Processor ($118.90 @ OutletPC)
CPU Cooler: CRYORIG - M9i 48.4 CFM CPU Cooler ($19.88 @ OutletPC)
Motherboard: ASRock - Z370M Pro4 Micro ATX LGA1151 Motherboard ($125.89 @ OutletPC)
Memory: G.Skill - Ripjaws V Series 8GB (2 x 4GB) DDR4-3000 Memory ($101.99 @ Newegg)
Storage: Western Digital - Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($46.55 @ OutletPC)
Power Supply: Corsair - CXM (2015) 450W 80+ Bronze Certified Semi-Modular ATX Power Supply ($46.99 @ Newegg)
Total: $460.20
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-02-07 12:31 EST-0500

New cpu, motherboard and ram will do good

PCPartPicker part list / Price breakdown by merchant

CPU: Intel - Core i3-8100 3.6GHz Quad-Core Processor ($118.90 @ OutletPC)
Motherboard: MSI - Z370 GAMING PLUS ATX LGA1151 Motherboard ($114.89 @ OutletPC)
Memory: G.Skill - Ripjaws V Series 8GB (2 x 4GB) DDR4-2666 Memory ($87.99 @ Newegg)
Total: $321.78
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-02-07 12:28 EST-0500

Everything really except the gpu. Such an old psu is not wise to use again anymore since it will have lost a decent amount of it's max capable output over the years due to wear.

Your E5200 is a dual thread processor with a passmark rating of 1489 and a single thread rating of 1017.
Good for a decade ago, but not so much by today's standards.

A $90 G4600 has 4 threads and a passmark rating of 5218 and a single thread rating of 2057.
The single thread performance is what most games need.

$50 buys you a lga1151 motherboard, and expect to pay $90 for a 2 x 4gb DDR4 ram kit.

The nice thing about this approach is that lga1151 has many cpu upgrade options.

On your current cpu overheating issues, it is likely caused by your cooler mount.
At idle, you should see 10-15c. over ambient.
Sooner or later, one should refresh the cooler thermal paste.
----------------how to mount the stock Intel cooler--------------

The stock Intel cooler can be tricky to install.
A poor installation will result in higher cpu temperatures.
If properly mounted, you should expect temperatures at idle to be 10-15c. over ambient.

To mount the Intel stock cooler properly, place the motherboard on top of the foam or cardboard backing that was packed with the motherboard.
The stock cooler will come with paste pre applied, it looks like three grey strips.
The 4 push pins should come in the proper position for installation, that is with the pins rotated in the opposite direction of the arrow,(clockwise)
and pulled up as far as they can go.
Take the time to play with the pushpin mechanism until you know how they work.

Orient the 4 pins so that they are exactly over the motherboard holes.
If one is out of place, you will damage the pins which are delicate.
Push down on a DIAGONAL pair of pins at the same time. Then the other pair.

When you push down on the top black pins, it expands the white plastic pins to fix the cooler in place.

If you do them one at a time, you will not get the cooler on straight.
Lastly, look at the back of the motherboard to verify that all 4 pins are equally through the motherboard, and that the cooler is on firmly.
This last step must be done, which is why the motherboard should be out of the case to do the job. Or you need a case with a opening that lets you see the pins.
It is possible to mount the cooler with the motherboard mounted in the case, but you can then never be certain that the push pins are inserted properly
unless you can verify that the pins are through the motherboard and locked.

If you should need to remove the cooler, turn the pins counter clockwise to unlock them.
You will need to clean off the old paste and reapply new if you ever take the cooler off.
----------------------------------------------------------------------------------

1) Yes, but I do not know when. Read here: https://www.pcgamesn.com/intel-cpu-roadmap-2018
2) If you trust your psu, have warranty on it and it can deliver at least 350W you can keep it.
3) For gaming 4 physical cores and 8GB RAM Are recommended. Forget the Pentium, if you want a lasting upgrade. Look into a ryzen build.

1> Intel have not given any confirmation yet and everything else is speculation. And with the recent Spectre and Meltdown debacle, seems like they have got their hands tied for some time.
2> As you can see here... https://linustechtips.com/main/topic/631048-psu-tier-list-updated/ it is a very low quality unit. How long you want to trust it with your brand new hardware, should be your call. Change it as soon as you can.
3> The Ryzen 1200 is a recent gen. quad core like the 8100 and should last you longer than the Pentium. Also with a b350 board as i have listed in my previous post, you can OC it to considerably reduce the performance gap... http://cpu.userbenchmark.com/Compare/Intel-Core-i3-8100-vs-AMD-Ryzen-3-1200/3942vs3931