I do love a good Spice & Wolf question! The short answers are:
Eve was not trying to hurt the Church. The only revenge she desired was against the deceased merchant who had once bought her, which was the reason she was so motivated to earn profits. Her goal was to become wealthier than he had ever been, in order to prove he'd only been able to buy her due to luck, and that he could never afford her under normal circumstances.
The suicidal act which Lawrence was referring to was her plan to buy up a large number of furs before everyone else, because it was purposely interfering with the expectations of the Church, and the Church isn't shy about killing people who defy their authority or otherwise become a nuisance.
Now for a bit of a longer explanation, specifically about the relationship between Eve's plan and the Church. While her plan wasn't a plot to get revenge against the Church, it might have seemed that way because she was determined to carry it out despite knowing it directly conflicted with the Church's plans. At the same time, the bishop of Lenos's unfair dismissal of Eve made the Church well-deserving of any negative impact her plans would have on them.
When the events are considered chronologically, it becomes clear that Eve didn't devise the plan to buy furs in order to seek revenge against the Church, since the way they wronged her in the first place was by abandoning her after she'd already shared this plan with them.
Basically, Eve wasn't interested in hurting the Church, but she also continued her plan in spite of the fact she would be hurting the Church. After this clarification has been made, your first question requires a new answer.
The Church is somehow hurt by Eve buying a lot of furs and then sailing away to make a big profit. How are these two things related?
For maximum clarity, I'll try to explain it with as much relevant background detail as possible.
Eve and the Church had been working together, smuggling salt into Lenos. Before this arrangement, the bishop of Lenos had been falling continually further into debt, until Eve approached him with the salt-smuggling plan. Since she's nobility within the kingdom of Winfiel, she also offered to put him in contact with a powerful archbishop there.
Eve did all the work in coming up with the plan, initiating the setup, then actually transporting the salt, and the Church paid her for its delivery. This arrangement was incredibly lucrative for the Church.
However, the Church was forced to cancel their annual northern campaign due to a falling out between them and the nation of Ploania, which is an area the campaign would have needed to pass through. Since the northern campaign's entire purpose had always been to display the Church's power, this cancellation brought the Church's authority into question and made the threat of an uprising more serious, so they started focusing entirely on strengthening their base of power and completely pulled out of salt-smuggling in the process.
This put Eve in a bad position, since she suddenly lost her sole source of income.
Meanwhile in the port town of Lenos, a freeze on all fur trading was put into effect.
(Note: The following section explains why a freeze was put on the fur trade, and why the Council of Fifty came to the decision that they did. If you fully understand this part already, feel free to skip it.)
The freeze on fur trading became necessary due to the northern campaign's cancellation. Lenos's craftsmen relied heavily on the northern campaign to sell their finished products, which would have normally flown off the shelves as souvenirs, since knights and mercenaries spent money quite freely. The cancellation of the campaign was an unthinkable economic blow to these craftsmen.
Since the campaign was not taking place, the town's economy would need to rely on merchants, who were not coming to the town to spend money as consumers, but rather quite the opposite. While knights and mercenaries are especially free with their coin, merchants are especially miserly. Their only purpose is to buy items they can resell for a worthwhile profit, so they have zero interest in buying clothing at retail prices.
Instead, the merchants would be interested in buying the furs themselves. As raw materials, they are cheap and can be easily sold for a good profit after transporting them elsewhere.
This is where the conflict arises.
With Lenos's craftsmen being unable to sell their products like usual, they'd also be unable to buy the vast quantities of fur like usual, which meant the merchants would have an opportunity to buy up the ridiculously large fur surplus that was suddenly available.
Additionally, the merchants could broker deals with the fur sellers, making arrangements to buy all their furs in the future as well. This would be pretty tempting to the sellers, since a merchant would be guaranteed to buy their furs every year, whereas the craftsmen of Lenos were now unreliable since the northern campaign might be cancelled again.
Thus, the Council of Fifty put a freeze on all fur trading, and convened to decide if the fur trade should be banned altogether, since that would guarantee the supply of furs would remain available to the local craftsmen.
The craftsmen of clothing in Lenos, along with the people who supply them their tools and wares, would be faced with complete ruin if the entire fur supply was bought out. At the same time, there was absolutely no guarantee that the clothing would sell even if fur sales were banned, and having money no longer coming into the town would devastate the economy of Lenos. Even if the craftsmen wanted to export the clothing, there were any number of other towns with superior clothing craftsmanship, so paying to ship it somewhere else would hardly be worth the trouble.
In the end, the compromise the Council of Fifty made was to restrict all fur-trading to cash-only transactions. By restricting fur trades to cash, they would be able to sell some of the furs while preventing the entire town's supply from being quickly bought out. After all, the larger a trading firm became, the more of its business took place on paper, in entries on ledgers, rather than with cash.
The Church heard of the decision well before it was going to be made public, and Eve found out through her contacts in the Church. She then approached the bishop of Lenos with a plan that would make both her and the Church a lot of money: Since the Church was sitting on a nearly unimaginable amount of cash from the tithes it collected, they could prepare to buy up all the furs in Lenos immediately after the Council of Fifty's decision was made public, while everyone else would still be scrambling to get cash together, and then they could move the furs downriver.
The bishop loved Eve's idea, except for the part where she was included in it. He found a trading company to partner with instead, and used that as an excuse to cut his ties with Eve, saying it would be more advantageous to deal with a trading company than with an individual merchant. It was a pretty harsh course of action, especially considering he owed a lot to her for the salt-smuggling opportunity. Although, the fact he owed her was exactly why he didn't want her around anymore, and got rid of her while he had a good opportunity.
However, Eve refused to let the deal she'd proposed get away from her. She started gathering up cash of her own, intending to buy a large number of furs and then transporting them downriver before anyone else had the chance, including the trading company the Church had partnered with. Whoever could get their furs downriver first would earn the best return on their investment, since people won't be willing to pay nearly as much for furs after they realize that the market is being flooded with them.
Eve's plan would infringe on a considerable portion of the Church's intended revenue, which is how it would hurt the Church.
Source: Spice & Wolf light novels (Volume 5).