How safe is a player-owned structure in high security space?

PoSes are safe in HS. They cannot be attacked without an active war, or the attacker will get CONCORDed. It is not completely safe, but if you see a notification for a war pop up, run over and put more hardeners/ECM on it. Don't bother trying to have the PoS attack back - you want to make it as annoying and difficult as possible for attackers to damage your PoS. Also remember to load it to the gills with Strontium, so that the reinforce timer will kick in.

The only things you can't put on a PoS in hisec are reactors and moon miners, IIRC. ( and of course silly things like jump bridges and capital things ).

The maximum level of security you can put a PoS into is 0.7, no matter how high your corp's faction standing is. The difference between 0.7, 0.6 and 0.5 is minimal - the biggest thing is the response time of CONCORD, something a PoS isn't terribly concerned with.

TLDR - War decs means that even high-sec structures are not truly safe and are still subject to extortion from powerful groups.

Like Phil Zitt says high sec POSes are generally safe, but they can still be used by bandit gangs to exhort money from you. A War Declaration (or War Dec in eve slang) allows a corporate to legally attack another corporation in high security space (high-sec) for an extendable period of time.

This means a powerful group can say "You better give me x cash or we're going to destroy your high-sec base and harass your comrades". Whether or not such would be bandits are serious or scamming is usually not clear. Some players just get bored and War dec a vulnerable group.

I've read or chatted with many people who have accounts of bandit corporations that go around looking for targets to threaten a war dec on in exchange for cash. Some of these groups even blog about their experiences.

Is it a good idea to give in to POS extortion?

Sometimes it's a good idea to pay them, sometimes it's better to ignore them, depending on their relative strength and how much harassment you can endure.

In addition to the other answers, the security of the system involved and closeness to trade hubs is significant. It is relatively easy to get to 5 standing. It is hard to get to 6 and a pain to get to 7.

Is the safety of the POS further influenced by anything else (ie: putting it in a 0.9 system instead of a 0.5 system)?

The highest security system that the game lets you anchor a POS is in 0.7. There are a number of people with 2 corps: one to wardec weak looking targets near trade hubs, and a real-estate corp that sells locations near trade hubs. These people will not bother folks in 0.7 systems.

Because there are more corps with 5 standing than moon slots near trade hubs, it is possible that your business competitors may decide to beat you up just to get a moon slot near the market.

There are 2 theories/doctrines of how to set up your POS: deathstar and dickstar. A deathstar is full of guns and will blast anyone who wardecs you (in null or low security space, they'll just start shooting because wardecs are not needed). In nullsec, most deathstars are set to NBSI ("not blue, shoot it", meaning shoot at everyone who comes in range who is neutral or bad standings). A dickstar is one with 40-120 ECM and electronic warfare modules (a large tower can put 40 online). In highsec, a dickstar is going to be a miserable target to wardec due to the jamming (since capital ships - which are immune to jamming - are banned in highsec). Judicious placement of secure containers when wardecs are made will frustrate the wardeccing corporation.