The short answer to both questions is yes, it is possible. Although as mentioned, both strategies have their disadvantages. A missile only strategy in particular is extremely expensive and may not be sustainable over a long term battle. See Steve's latest AAR for an example of the cost of missiles even as part of a strategy, he had multiple situations where a missile strike was an important element of the attack, but also cost about as much as a full size warship to pull off.
On the other hand beams need a lot of work too, speed is a major issue of course, anything that can outspeed and outrange you can kill you without risk, you can mitigate that a bit with a fighter strategy, since they tend to be the quickest things in space and relatively cheap. Missile defense is doable with a beam only setup but you will need a high volume of fire to make it work, that is the same for anything though I suppose. Logistically they are much easier than missiles too, but that isn't to say there are no considerations, the latest changes to failure rates mean that MSP is something you'll need to pay close attention to during ship design and monitor closely in combat, particularly in extended fights or in situations where you might have multiple fights in succession.
Don't let the issues stop you though, it's all doable for sure and if you wanna run an empire in a particular way for RP or just because you like one strategy better then go nuts, you will still be able to enjoy the game.