"很高兴认识你" or "我也很高兴认识你." There must be a simpler way to say this in Chinese. 

Believe it or not, the most common way of saying "nice to meet you" is 很高興認識你 or 認識你很高興 (I myself favour the order of the second one). 認識 implies getting to know someone pretty well. If you just chat to someone for five minutes, it's more common to say 很高興見到你. It doesn't sound awkward to native speakers. 

You will continue to make the most basic of mistakes for the longest time, even the most advanced speakers will get the most basic things wrong from time to time. The best thing you can do is just keep trying your best. 

I couldn't recall the words for basic phrases like "How do you say X in Chinese?", "Can you please repeat that?" and my new favorite "Can you please try saying that in a different way?"

The more you speak to people in social situations, the better you will get at saying these things. If I had to say these, I'd say:

XXX用中文怎麼講？

你能再說一遍嘛？

你能換個說法嘛？

The last thing that really struck me from the meeting was how even the slightest pronunciation error could make an entire sentence unintelligible. (A number of off-by-one errors.) A few examples:

This is debatable. I have had many heated conversations with many different people about this. The conclusion I myself have come to is that small mistakes can cause misunderstandings or lead to the person you are speaking to not understanding, but so long as there is enough context, you'll probably be understood. The only time you have to be really careful is when giving one word answers. 

1) I was trying to say, "我用三脚架固定“ (I use a tripod to fix/stabilize my phone (for filming). But instead of saying sān jiǎojià, I said sānjiǎojiù. 董老师 had no idea what I was saying.

Usually pronunciation mistakes concern tones, not getting the sounds wrong. When you get the actual sound wrong, it will be harder for the native to understand. For example, if I said "though" instead of "through", a native might not understand me. Also, I myself would say 我用三腳架架住相機 myself. I think that's the way it's said, and also by using more words it's clearer what I want to get at. 

2) I tried to say, "我每天都学习，每天都进步." (Every day I study, every day I improve.) But I said jìnjù instead of jìnbù. Again 董老师 had no clue what I was saying.

See above

3) One of the attendees introduced himself with his Chinese name. The first syllable was lan, but he pronounced it làn instead of lán (蓝). I was trying to guess the meaning of his name, but it didn't occur to mean that the first character was actually 蓝 and he had just mispronounced it.

You yourself will also have to get used to people getting the tones wrong too, especially if you interact with foreigners who speka Chinese. 

I wonder how much this holds true in Chinese where there is much less exposure to non-native speakers. Would more context make me easier to understand? And am I going to have a harder time talking to native speakers in China than the Chinese teachers who live here in the US? 

Some people are better at understanding you than others, and some people when they get to know you more will just be able to tell what you are trying to get at. Try speaking to them more. Yes, you will find speaking to people here in China much more difficult, as they are many people who don't speak clearly. A teacher of Chinese in the US will probably speak clearly.