Many of use Castrol 5W-30 from the dealer, and the OEM filter, because they are not expensive, and the quality is fine.

Agree entirely with your plan to do a first oil change on your new-to-you vehicle. After that, however, I wouldn't do anything at 5000 miles, unless a year was up. That is 3 times as often as the pictures showed, and they showed the results over a much longer test cycle. You are out of the free maintenance period, so it is really up to you. People only opt for intervals of 7500 miles because it is an interim change between the dealer-supplied 15,000 mile changes (assuming high annual mileage). You could run it to 10,000 miles without any worries at all, you would be 5,000 miles in front of the factory recommendation, which was made on a poorer oil than is available today (check the API specs for your 2001 model in your manual, and what is sold now at the dealer, such as API SL).

I just don't see the point of checking the filter at 5000 miles. If you are going to open it up, just replace it even if you don't change the oil. Or you could just leave it.

Just my $0.02.

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2007 X3 3.0si, 6 MT, Premium, White

Retired:
2008 535i, 6 MT, M Sport, Premium, Space Grey
2003 X5 3.0 Steptronic, Premium, Titanium Silver
2002 325xi 5 MT, Steel Grey
2004 Z4 3.0 Premium, Sport, SMG, Maldives Blue