1. Is there a guideline for value in Gold vs tech trades? Tech for tech is easy to just compare beakers, but when trading gold are there rules of thumb or is it just whatever it is worth to you?
A good general rule of thumb is that gold is more valuable than beakers, because beakers have research multipliers. For instance, 300 gold is more valuable than a 300 beaker tech. The exception is if you urgently need the tech, in which case it may be worth paying the amount of beakers in gold.
If you're buying a tech from the AI, pay the minimum that they'll accept (put the tech on the table and ask "What do you want for this?"). Be warned that the AI's will frequently massively overvalue their techs, so check what you're paying for before you accept the deal.
If you're selling a tech to the AI, obviously get the maximum amount they'll give to you. Be warned that when the AI doesn't have much money, they'll make really raw deals (eg Civil Service for 30 gold). Don't accept unless it's an amount you're willing to part with the tech for. Note that if you're much more advanced than the AI, it's sometimes worth selling them old techs for very little gold to boost your treasury.
Between human players in multiplayer, you'll have to work it out between yourselves, but generally most players will think a 2:3 or 3:4 ratio is roughly fair (ie, spending 200 gold to buy a 300 beaker tech, or 300 gold to buy a 400 beaker tech). Again, the rule with technological advancement applies - if one player is much more advanced than another, they will be expected to sell their old techs for much less than their worth in beakers.
2. Barbarians spawning in "clear" areas. I take effort to clear as wide a circle as possible around my cities. I notice barbatians tend to just pop up a tile or 2 away from my borders instead of coming from dark areas a lot of times. Is this supposed to happen?
Barbarians can appear in any tile which you (or another player) cannot see into. They can (and do) spawn far from your borders, but you don't notice these barbarians because you don't see them. You only see them once they travel about and finally reach your borders. I suggest starting a new game and checking WorldBuilder every 10 turns or so, to confirm this for yourself.
By the way, if you're having trouble with barbarians appearing near your borders, one of the best counters is to position several units outside your borders (especially on hills, as they get greater visibility) and keep as many tiles as possible "un-fogged". In this way you'll catch barbarians long before they get close to your borders.
