But really? Should this programme take so long, 5 days already plus another 8 days just to download it??
Yes. That is normal.
Bitcoin Core requires the entire blockchain, which currently takes up ~170 GB of disk space. The entire blockchain must be downloaded and validated. This means that you need to have a fast internet connection and a fast processor in order to be able to download the blockchain and process blocks quickly. Furthermore, while validating blocks, Bitcoin Core will need to read many things from the hard drive, and write many things to the hard drive. So a slow hard drive is going to make the sync take even longer.
There are likely 3 things that are causing you to have a slow sync time. Firstly, your laptop has a mechanical hard drive (that's what HDD means). A mechanical hard rive has moving parts and because it has moving parts, it is slow. Thus this is a bottleneck that will slow down your sync time because it takes longer to read and write data to and from the mechanical hard drive.
Secondly, you are using a laptop, which means that you are likely using a WiFi connection. WiFi can be slow. It depends a lot on any interference, the materials between your laptop and the router, and the router that you are using. It's likely that your WiFi connection is not all that fast and thus syncing will take a longer time. Also, WiFi means that there can be more data loss and corruption, so it will take longer for the networking protocols to either resend the data or recover it.
Lastly, you are using a laptop, which means that you are likely using a slow processor. Generally Intel Core i5's are decent, but it depends on what kind of processor you have. Many laptop processors, even if they are Core i5 or Core i7, are very low powered because laptops need to have lower powered components in order to have a better battery life. This means that your laptop could be running a low powered processor which is slowing down the sync as it cannot process blocks quickly.
Does it take everyone this long to get started?
It does for everyone using Bitcoin Core. It really depends on the computer that you are using. For some people, they can be synced in 6 hours. For others, it takes weeks. But for everyone who uses Bitcoin Core, there is always a period of waiting for it to sync when you first get started.
This is also recommended to be installed to a flashdrive or hard drive away from your PC in the event of a malfunction.
No, don't do that. It absolutely is not recommended to do that. The USB protocol and the USB connection is not very fast and not very good for transferring large amounts of data without some corruption. Storing the Bitcoin Core data on an external drive is a good way to have it corrupted. There have been many more cases of people's Bitcoin Core data directories being corrupted routinely when using an external drive than when using an internal drive.
2. There is a number of web wallets that can accomplish what the Core does in an acceptable way. BlockChain for instance is the number one website and the original web wallet i'd say. It offers a wallet ID that you could save or print, and you can extract your private address or access on any other application or website that allows a private key upload. You can find a number more by googling "Bitcoin Web Wallet" and ensuring you do your research on each website that pops up.
Don't use web wallets. If you want to use a wallet that does not have to do this kind of syncing, use a wallet like Electrum. Don't use web wallets.