With the introduction of radio in warfare, was there any form of arms race between the combatants over it? I understand there was interception through units like the British Room 40, but was there any attempts at things like jamming or using radio communications in deceptive ways (like sending false signals)?
Electronic warfare predates radio. The US Civil War saw early electronic warfare, such as destroying telegraph lines to cut communications, and the tapping of telegraph lines.
Early WW1 electronic warfare was along similar lines, with the British cutting the German undersea telegraph cables that passed through the English channel on the 5th August 1914 (early in the morning on the day after Britain declared war on Germany). This was pre-planned, as early as 1912. German telegraph traffic under the Mediterranean used British cables, so was stopped without any cable-cutting. This left a single German undersea cable operational, from West Africa to South America (and West Africa was telegraphically isolated from Germany). This forced Germany to use radio to communicate with their colonies and for any other overseas communications, which gave the British a wealth of radio intercepts to flex their codebreaking skills on.
The Germans retaliated by attacking cable stations on islands in the Pacific and Indian oceans, and also attacking wireless stations.
Other than that, most WW1 electronic warfare was the interception of signals, including signal traffic analysis, whereby information about enemy forces is deduced from the quantity and source of radio traffic, without codebreaking. Battlefield telephone systems were listened to, either by directly tapping the wires, or using induction to pick up signals from nearby phone lines.
There was some use of deception, with signals transmitted by the enemy pretending to be friendly. One very successful example was the signal that resulted in the zeppelin L59 abandoning its supply mission to German East Africa. L59 was carrying about 15 tons of supplies, and had crossed the Mediterranean. When it was near Khartoum (Sudan), it received a signal, from the British but supposedly from Berlin, informing it that the German forces in German East Africa had surrendered. L59 abandoned its mission, and returned to Europe. (The Germans in East Africa only surrendered after the war in Europe was over.)
Jamming didn't play a large part in electronic warfare at the time. Partly, this was due to the early radio sets being spark-gap sets, the reliable jamming of which would have required jamming transmitters of much greater power than the radio set transmitting the signal. In the absence of electronic amplifiers early in the war, having a transmitter of much greater power was a rather difficult task. Jamming had been consider even before WW1, with a jamming mission proposed by the Russian navy in the lead-up to the Battle of Tsushima. A Japanese auxiliary cruiser was transmitting information about the location of the Russian fleet to Japan, and the captain of one the Russian ships requested permission to attempt to jam the transmission by simultaneously transmitting a signal on the same frequency. The Russian admiral refused permission, and we don't know whether the attempt would have succeeded.