I'm curious as to what everyone else thinks about advisors. Are the suggested spots for founding cities, researching techs, and producing things in cities good pieces of advice from the game's advisors?
Not really. The only times the advisors seem to get it right is in the same cases a novice player would, that remains sound advice throughout the game. They can't take account of context - a city is always desirable next to a luxury however many of that luxury you have, regardless of where it is in relation to other cities/civs, their advice for a building won't take account of your plans for a specific city or whether you have enough of those buildings elsewhere in your empire, etc.
I'm also asking this because I have been having issues winning regular games on the third difficulty level (Warlord?). I tend to follow the advice given by the advisors, because I don't really have a developed strategy of my own. Is this a good idea or not?
No, it's a good idea to get a feel for why you'd build that there/then, research that tech at that point etc. Only follow an advisor's advice if you can see why it's a good idea ... in which case you'd normally go for it anyway.
Are Declarations of Friendship worth the diplomatic repercussions? I don't usually accept friendship requests because I don't want to make any enemies. However, it seems like the AI likes to declare war on me anyway, destroy 1 or 2 of my units (and I destroy one or two of theirs), then ask for peace (pointless war). Not sure if Declarations of Friendship would help with that or not.
As you get to higher difficulties and war becomes difficult to win, you will find that a lot of the game's diplomacy revolves around declarations of friendship. You don't want to make friends indiscriminately, and over time you'll find that declarations are less likely to be renewed or have less lasting effect with some leaders than others. You'll also get a feel for who's aggressive, and making friends with them can prompt them to attack someone else - admittedly at the expense of upsetting the other guy. Civ V diplomacy tends to play out as the development of 'power blocs' that face off against each other - a few civs with mutual friendships, and if a civ is a friend of your friend, you're more likely to get friendship with them. The identity of powers in a bloc isn't always stable, but you'll want to consider which bloc you'd rather be a part of - not just whether Civ X wants to be your friend, but whether you want to be friends with Civs Y and Z as well. If you make friends with India, say, but India is friends with Japan, you can be fairly sure that at some point Japan is going to declare war on the other guys. To maintain good relations within your bloc, you'll probably find it helpful to do the same (usually, unless you do something to keep your friends happy, they'll be increasingly less inclined to renew DoFs in future).
One example of a case where DoFs can be particularly important is India. India likes to play diplomatically, by building up a big network of friends. Then, they will usually denounce somebody - as a result of which all their friends will start denouncing the same civ, and sooner or later someone will declare war. You will usually want to be India's friend in those kinds of situation, unless it's weak in that game (hasn't made many DoFs).
Advisers are useful for telling who has a bigger military and who has a smaller military. This is information you can't get just from looking at the demographics screen (only the best, worst, and average)
This is somewhat useful, but only so long as you bear in mind that this seems to be based on a straight unit count - I've had civs that could "wipe me off the face of the planet" with giant armies of warriors when I've got several Swordsmen and catapults. And as for the 'who's winning the war' bits, I can't even work out how that's programmed - I've had wars where I haven't lost a unit, the enemy's lost a dozen, and I'm in their territory - and "the war is not going well".