Full node or Full node third-party services pros and cons I'm setting up P2P Bitcoin website similar to LocalBitcoin for school project.
Before I go deep into this journey I'm doing some plannings.

I have lot's of questions in my head and have been studying bitcoin technology.
I would like know if its better to run full node on my server and execute RPC commands or should I just look for companies that offer such services to save me some time?

What are the  pros and cons of running full node
I will also appreciate recommendations for third-party services for Bitcoin P2P market
What are the pros and cons of running full node


it all comes down to the level of trust. Whom do you trust? Your own system, or someone else? And then, how much money are we talking about? Several low values, or monthly or yearly values?

For sure I recommend a full node. And yes, it has the disadvantage of downloading the blockchain (easily up to three weeks), and you need to have that infrastructure up and running. For this you gain the absolute security, that noone faked a block or a tx in the blockchain, that is assembled on your system. Also the level of privacy is higher than e.g. SPV nodes. A full node supports the decentralization aspects of the network. More full nodes, more tamperproof is the whole bitcoin eco-system. 
Using RPC calls to your local bitcoin node requires then some developer knowledge, or a dev-team. That can add to costs in setting up your environment. 

Using services like bitpay or similiar has the advantage of getting quickly started. This way you can start to setup your business, and later on, if you have really big values, you are still able to convert to a full node. 

So from a risk perspective: when it is low values, go ahead with services from third parties, but when you achieve higher values (above monthly income), think about securing your environment with an infrastructure, that is under your own control.