So on my upcoming flight I will be flying AA into LHR and then flying BA on Aer Lingus metal to Shannon. Since I have Plat with AA I should get a free checked bag with BA. Does this also apply to BA marketed flights? Will I get a free checked bag on my flight to Ireland?

Also I have heard that the carry on rules are much stricter in Europe than in the US. I an used to taking my 17" laptop backpack as my personal item as well as my 21" roller bag. I assume this will not fly with BA?