Do my MAs change over time or they stay the same?

Example: H1 MA21 is a sum of last 21 prices that hourly bars closed with divided by 21. So if the closing prices were not changed the MA reading for any bar should not change.
Accuracy depends on the platform you use for charting. MT4 , great for custom indicator, but charting is a bit off. Especially when testing with the new broker.
Have in mind that if the broker is new you can not test your strategy on the prices quoted by the broker.Only the candles that were drawn during their operation are actually the prices they were quoting to their clients.
The rest of data is historical data taken from MetaQuotes most probably. MQ data gets easily corrupt when they change mt4 builds so be aware.
If you trading rely on data accuracy look somewhere else.