applying thermal paste after cleaning the heatsink

It's not going to cost you badly, just go and get some thermal paste like ic diamond and slap on a pea size amount and whatever

Keep a temp monitoring program open just to watch the CPU temps. IF the temps stay good then you will be OK for a few days but keep an eye on them just in case.

Tech Tip: to clean your CPU and cooler use rubbing alcohol, as close to 100% as your local store has and coffee filters. Rubbing alcohol evaporates cleanly and coffee filters do not leave fibers behind.

Use alcohol and coffee filters to clean off old paste. Coffee filters are lint free.
Arctic Silver is very popular as thermal paste.

Plenty of videos on youtube show you how to clean the old cpu/heatsink and then to apply paste. Watch a few in case you hit someone who does not know what they are doing.

Do ground yourself by touching power supply before touching heatsink and again before touching cpu, then again touch PSU after you clean the parts right before you re-install them. Static can kill your CPU.

Before you touch anything inside the case touch something metal to discharge your body of static electricity. The static charge in your body can hurt the components inside the case. They also sell a wrist band that has an alligator clip on the the end of a wire to keep your body grounded but the cheap way is to touch the case/metal before you touch anything inside the case.

This one at least is simple, unlike many pc things. The power supply (also called PSU) is the box that the wall line plugs into and is the source for all power in the PC. The outer metal of the PSU is well grounded. (The inside of a PSU is never safe -- there can be voltage there even when unplugged because of capacitors, never open a PSU) . Before you add or delete any card (like a video card) or memory reach over and touch the PSU. That grounds you relative to the PSU. It means there will not be a spark between the card or memory you are inserting and the ground provided by the PSU. This can save your part from damage when you install it.

The scenario you are trying to avoid by touching the power supply:: You walk over to the PC. It is a dry day and you have a rug, By skuffing your sneakers over the run you develop a static electric charge. You pass this to the card or memory dimm you are installing because you are touching the card or memory dimm. When you install the card or dimm the voltage discharges to ground through the card or memory dimm destroying the card or dimm. This happens often enough that people who update PCs for a living wear a "ground clip" the is a wire that ground them. You don't need to go that far, by touching the PSU you can remove the low current/high voltage static charge and protect your card or dimm.

Here is an example of a ground clip. https://www.newegg.com/Product/Product.aspx?Item=9SIA8HV56Y1275 you dont need one, just touch the PSU.

Actually the PSU is not grounded unless the power cord is plugged, Not advisable to have your hands in the case when the PSU is plugged in.

The Outer case of the PSU is actually mounted to the case making all metal surfaces of the have the same grounding property's as the PSU, SO and metal surface of the PC and case will discharge the static. Even if the chair you are sitting in has metal this too will discharge the static. So any metal surface will suffice.

Static can fry more than the GPU or Ram. It can kill any piece of electronics, motherboard or more specifically most components on it (capacitors can handle the charge), It can fry the ODD if it has a circuit board exposed, HDD's and SSD's if any of the circuits are exposed so be sure not to touch the power or sata connections on any components with out first being grounded, the CPU is extremely sensitive to voltage and the motherboard can carry the static shock to the CPU so even if you do not touch the CPU it can still be hurt by the shock.

I have a board right now that got zapped a few years back while in storage and now the BIOS will not hold the settings once the power cord is removed, Yes I did change the battery and it still will not hold the settings. While this is a just an old Core 2 system the point remains the same, Discharge yourself by any means you can before reaching into the case. any time you take your hands out of the case you need to touch it to be sure all static has been discharged. I personally purchased a anti-static strap because I do so much with PC's I can not take the time or worry about constantly trying to discharge myself.

While working on your PC be conscious of not rubbing your feet on the ground, (as mentioned before) scuffing your feet when you walk around it or to get a part, if sitting in a chair try not to wiggle your bottom in the chair seat as this will generate static.

Hopefully we have given you more than enough reasons to discharge yourself and things to mindful off so you do not damage your system. Good luck, Have fun while working on your system.

Yes, that's why we talk about relative ground "...and touch the PSU. That grounds you relative to the PSU..." The voltage difference we care about is the one that causes a spark between your part and the MB which is also grounded relative to the PSU.