It is unlikely to be a battery issue. The hydraulic spoiler has microswitches that cut in at 75 and 35mph to raise and lower it. These (one for each function) may need replacing - the cheapest test and solution.
If not microswitches, the next most common is leaking rams - lift the bootlid and see if there is fluid on the black plastic tubes on either side Those are the rams. If its this, then you can mess around with a DIY fix with new seals off eBay and refilling them. Alternatively, source a scrap set of rams with wiring loom from a scrap yard. The alternative is a new spoiler mechanism from OPC - will be around £2k fitted.
These cars are aging well when you consider it is on its factory spoiler and rams till now. I'd try the microswitches first. Its relative easy DIY and take it from there. A good Indy should be able to sort it either with a scrap replacement or a replacement setup from OPC. Indys may either suggest a repair or a replace solution for the spoiler malfunction given the labour considerations of trying a repair and an unpredictability of if the solution will work or hold for how long.
The oil light might be a faulty sensor. The oil pressure drop and then comes back up. Check the oil level though. Non Mezger cars such as 996 4S have a wet sump (or as Porsche describe - 'integrated dry sump' - and if its low on oil, when going around corners or roundabouts the low level may make the sensor 'dry' triggering the light.
If car starts OK and horn and lights are fine, the battery is probably OK.
An Indy can read if any fault codes are triggered for the spoiler and the battery and oil lights and having an Indy look into it is probably the solution I'd take.