1) When trade via MT4 terminal you always will see a bit delay. The reason is that all trades passes via MT4 trade server - as fxmbs said in his post. It does not matter of the order type, all of them passes for execution first in MT4 server and this cannot be avoided (if you trade with MT4).
Trading via FIX API connection could make the execution faster and this needs to be connected to LMAX environment, but avoiding MT4 at all. (just have in mind, LMAX does not like news/spike trades and arbitrage traders)
2) With LMAX you may get average price. I mean when you send order for example 500K, but the Top of the Book is just 100K, in order to execute your request of 500K, the first 100K are executed on the price you see, then the next sitting orders on different price /next sitting prices and etc. until your order is filled. In overall you see executed order with slippage, but this in fact is most likely averaged price of all executed orders up to your request of 500K - and this is depending from the market depth of the traded symbol.
3) The reason to see execution of stop loss with slippage could be the same or similar as described in point 2.
I am not sure whether is possible their MT4 accounts to have related LMAX platforms accounts where you may compare the trades, but with sure if you have LMAX account you will see the partial execution of the larger orders.
Have in mind LMAX is electronic exchange and your order often matches for execution other retail client's orders in the ECN, which most likely could explain the point 2 and 3, because LMAX is not just a broker like others either to send orders for execution in third party (ECN/STP) or MM broker.
I hope this will help.