Any DNS server still needs to lookup addresses, but if it's configured to keep a cache of stored addresses you may lower the number of times that a certain address needs to be looked up outside of your own network. You're not guaranteed to see a real benefit from this as a single user, though.
However, running a local DNS can give you additional benefits, though: There are products like Piehole that block some scripts and ads even before they get a chance to load, by knowing beforehand which domains are mostly used for that kind of crap. Using that will definitely speed up your Internet experience.
Different browsers are good at different things. Some browsers do one thing really well, and if you mainly do that on the web, using that particular browser will improve performance.
Rendering modern web pages is an expensive process, computationally. A fast computer will do that faster than a slow one, simplistically speaking.
Yes, because TCP - the protocol used underneath the web page - is a two-way communication. A request for an element may look like this:
(C=client, S=server)
C: Hello?
S: Hello.
C: May I look at document "sexycatpicture.jpg"?
S: You may look at document "sexycatpicture.jpg".
C: OK, so please send me document "sexycatpicture.jpg".
S: Getting ready to send document "sexycatpicture.jpg" which is 54 KB in size. Are you ready to receive "sexycatpicture.jpg" which is 54 KB in size?
C: Yes, I'm ready to receive "sexycatpicture.jpg" which is 54 KB in size.
S: Sending.
C: Receiving.
S: I have sent "sexycatpictyre.jpg" which is 54 KB in size. Did you receive "sexycatpicture.jpg" which is 54 KB in size?
C: I have received "sexycatpicture.jpg which is 54 KB in size. Thank you.
S: De nada.
C: OK, bye!
S: Bye!
Now imagine that each part of this interaction requires 2 ms to reach its target, or 150 ms to reach its target. You will notice a 75 times decrease in performance. Add to this that sometimes you're not alone in wanting to watch "sexycatpicture.jpg", so the server may struggle to deliver on your request immediately. Or the picture has been archived and needs to be found on old-and-slow mechanical storage. Or any of a number of various things that affect immediate performance beyond just network latency.