Nope, no soldering.
Look at the iFixIt and OWC sites for the drive replacements and iFixIt for the battery. As you replace the battery, you'll see that the drive is a side job that takes a few minutes extra.
Besides the list below, you'll need a bottle of white-out or light colored, opaque lacquer like nail polish. There will be a few tiny connectors on the motherboard and inline. Marking one side of each half will show you the correct polarity when reassembling. On a few, it's not that hard to force it back incorrectly causing problems like the fans going on full or the microphone not working. A little white paint saves tons of time.
As the battery gets weaker, it will cause settings to be lost but black screen and/or weird objects mimicking GPU problems are also common. Most of the older Macs that come to me for GPU problems leave with a new battery and nothing else. Yours is 10 years old, way, way past time. A CR2032 battery from the pharmacy or convenience store is fine. The BR2032 that Apple uses has a wider heat tolerance, that's all.
Ok. These links are to Amazon UK.
Tons of SSDs on the market. Everyone I know is using the WD Blue 3D or Sandisk Ultra 3D (same drive), Samsung 860 EVO or Crucial MX500. All have 3D Nand and 5 year warranties. The SanDisk 3D has it on price at the moment 250GB–4TB with 3 sizes in between.
https://www.amazon.co.uk/Western-Digital-WDS500G2B0A-Blue-Internal/dp/B073SB2MXW/ref=sr_1_1?keywords=wd+blue+3d&qid=1567198269&s=gateway&sr=8-1&th=1
You'll need the following magnetic Torx drivers: T6, T8 & T10, a Phillips #2. A pair of suction cups, a spudger or two, a tool that pokes all come in handy.
This kit should have the tools you need except for a Phillips driver and suction cups. Magnetized drivers are a must, IMO.
https://www.amazon.co.uk/Kaisi-Scre...VGAE44XADV98T5XR8441&qid=1567196715&s=gateway
There are many ways to get the front glass off but nothing beats a pair of suction cups like these—one each in the upper right and left hand corners and pull till the magnets release.
https://www.amazon.co.uk/MMOBIEL-Su...s=imac+tools&qid=1567195181&s=gateway&sr=8-38
You will need to deal with the temp sensor—remove it and the fans go full-on. The software so-called solutions absolutely suck. With the 2009-10, you have three choices (the 2011-on are different and the following won't work):
a) short the leads of the one in there already—keeps the fan quiet but if the drive warms up the fans don't come on. I really don't like that.
b) Use an optical drive sensor instead (not the one on your optical drive!). It works but isn't calibrated for a hard drive or SSD. Still, it works and is less expensive than (c)
https://www.amazon.co.uk/Replacemen...c+temp+sensor&qid=1567198965&s=gateway&sr=8-2
c) OWC has a sensor specifically for the late 2009-10 iMac.
https://www.amazon.co.uk/OWC-OWCDID...c+temp+sensor&qid=1567199707&s=gateway&sr=8-1
The right bracket is convenient and helps keep the drive cool. Or you can use double-stick foam tape to attach the drive to the back after you remove the HDD. Some stick the SSD to the old drive and leave the old sensor in but I don't recommend that—the internal cooling is worse and there's no heat sensor? Yikes!
Anyway, lots of bracket designs but this is the one for the iMac. If it looks like this, you're good.
https://www.amazon.co.uk/Sabrent-De...o+3.5+adapter&qid=1567200158&s=gateway&sr=8-3