user125553: I have a task where I need to set up some sort of a listener on a particular bitcoin addresses for incoming transactions, and once a transaction arrives, I need to call some function in my web service. Is it possible to have some sort of an event be triggered by the incoming bitcoin transaction that calls my web service? What technologies or tools are available that I could use for this task?
Furthermore, after the function in the web service has completed, I need to forward those received bitcoins to another address using that same web service. The web service can, if needed, be provided access to the private key for the address from which these bitcoins would be sent out. What tools/technologies could I use for this purpose?

user-1: 
I have a task where I need to set up some sort of a listener on a particular bitcoin addresses for incoming transactions, and once a transaction arrives, I need to call some function in my web service

If you are running Bitcoin Core, ZMQ can be used for listening to incoming transactions and blocks: https://bitcoindev.network/accessing-bitcoins-zeromq-interface/
This will provide you a live feed of new transactions added to mempool or blocks being mined. In the background you could process these and check if a Bitcoin address is used in any transaction.

Furthermore, after the function in the web service has completed, I need to forward those received bitcoins to another address using that same web service

If the above code returns any transaction in which address was used as output, you can use this information to create a transaction with createrawtransaction or createpsbt RPC. Sign transaction and broadcast it.
If you are looking for a library, it depends on the language used. Example: NBXplorer for .NET, JavaScript libraries

125553: <p>I have a task where I need to set up some sort of a listener on a particular bitcoin addresses for incoming transactions, and once a transaction arrives, I need to call some function in my web service. Is it possible to have some sort of an event be triggered by the incoming bitcoin transaction that calls my web service? What technologies or tools are available that I could use for this task?</p>
<p>Furthermore, after the function in the web service has completed, I need to forward those received bitcoins to another address using that same web service. The web service can, if needed, be provided access to the private key for the address from which these bitcoins would be sent out. What tools/technologies could I use for this purpose?</p>
 -1: <blockquote>
<p>I have a task where I need to set up some sort of a listener on a particular bitcoin addresses for incoming transactions, and once a transaction arrives, I need to call some function in my web service</p>
</blockquote>
<p>If you are running Bitcoin Core, ZMQ can be used for listening to incoming transactions and blocks: <a href="https://bitcoindev.network/accessing-bitcoins-zeromq-interface/" rel="nofollow noreferrer">https://bitcoindev.network/accessing-bitcoins-zeromq-interface/</a></p>
<p>This will provide you a live feed of new transactions added to mempool or blocks being mined. In the background you could process these and check if a Bitcoin address is used in any transaction.</p>
<blockquote>
<p>Furthermore, after the function in the web service has completed, I need to forward those received bitcoins to another address using that same web service</p>
</blockquote>
<p>If the above code returns any transaction in which address was used as output, you can use this information to create a transaction with <code>createrawtransaction</code> or <code>createpsbt</code> RPC. Sign transaction and broadcast it.</p>
<p>If you are looking for a library, it depends on the language used. Example: NBXplorer for .NET, <a href="https://bitcoin.stackexchange.com/questions/99935/the-best-javascript-libraries-supporting-bitcoin-core/">JavaScript</a> libraries</p>
